hcps geometry curriculum guide

Geometry Pacing Guide and Curriculum Reference Based on the 2009 Virginia Standards of Learning

2016-2017

Henrico County Public Schools

Henrico Curriculum Framework Geometry

Henrico County Public Schools Page 1 of 72

Introduction The Mathematics Curriculum Guide serves as a guide for teachers when planning instruction and assessments. It defines the content knowledge, skills, and

understandings that are measured by the Standards of Learning assessment. It provides additional guidance to teachers as they develop an instructional program

appropriate for their students. It also assists teachers in their lesson planning by identifying essential understandings, defining essential content knowledge, and

describing the intellectual skills students need to use. This Guide delineates in greater specificity the content that all teachers should teach and all students should

learn.

The format of the Curriculum Guide facilitates teacher planning by identifying the key concepts, knowledge, and skills that should be the focus of instruction for

each objective. The Curriculum Guide is divided by unit and ordered to match the established HCPS pacing. Each unit is divided into two parts: a one page unit

overview and a Teacher Notes and Resource section. The unit overview contains the suggested lessons for the unit and all the DOE curriculum framework

information including the related SOL(s), strands, Essential Knowledge and Skills, and Essential Understandings. The Teacher Notes and Resource section is

divided by Resources, Key Vocabulary, Essential Questions, Teacher Notes and Elaborations, Honors/AP Extensions, and Sample Instructional Strategies and

Activities. The purpose of each section is explained below.

Vertical Articulation: This section includes the foundational objectives and the future objectives correlated to each SOL.

Unit Overview:

Curriculum Information: This section includes the SOL and SOL Reporting Category, focus or topic, and pacing guidelines.

Essential Knowledge and Skills: Each objective is expanded in this section. What each student should know and be able to do in each objective is outlined.

This is not meant to be an exhaustive list nor is a list that limits what taught in the classroom. This section is helpful to teachers when planning classroom

assessments as it is a guide to the knowledge and skills that define the objective. (Taken from the Curriculum Framework)

Essential Understandings: This section delineates the key concepts, ideas and mathematical relationships that all students should grasp to demonstrate an

understanding of the objectives. (Taken from the Curriculum Framework)

Teacher Notes and Resources:

Resources: This section gives textbook resources, links to related Investigating Geometry Online (IGO) modules, and links to VDOE’s Enhanced Scope

and Sequence lessons.

Key Vocabulary: This section includes vocabulary that is key to the objective and many times the first introduction for the student to new concepts and

skills.

Essential Questions: This section explains what is meant to be the key knowledge and skills that define the standard.

Teacher Notes and Elaborations: This section includes background information for the teacher. It contains content that is necessary for teaching this

objective and may extend the teachers’ knowledge of the objective beyond the current grade level.

Extensions: This section provides content and suggestions to differentiate for honors/Pre-AP level classes.

Sample Instructional Strategies and Activities: This section provides suggestions for varying instructional techniques within the classroom.

Special thanks to Prince William County Public Schools for allowing information from their curriculum documents to be included in this document.



Geometry Pacing and Curriculum Guide

Course Outline First Marking Period at a Glance

Unit 1: Transformations (G.3cd)

Unit 2: Fundamentals (G.4abef)

Unit 3: Logic (G.1abcd)

Unit 4: Parallel and Perpendicular

Lines (G.2abc, G3ab)

Second Marking Period at a Glance

Unit 5: Triangle Fundamentals

(G.4, G.5, G.10)

Unit 6: Proofs with Congruent

Triangles (G.3a, G.6)

Unit 7: Similarity (G.7, G.14a)

Unit 8: Right Triangles (G.3a, G.8)

Third Marking Period at a Glance

Unit 9: Polygons (G.4, G.10)

Unit 10: Quadrilaterals (G.3a, G.9)

Unit 11: Circles (G.11ac)

Unit 12: Equation of Circles (G.12)

Fourth Marking Period at a Glance

Unit 13: Surface Area and

Volume (G.13, G.14bc)

SOL Review

View the online HCPS Pacing Guide for more details

Big Ideas

1. Logic & Reasoning 2. Parallel Lines 3. Properties of Triangles 4. Polygons 5. Proportional Reasoning

6. Quadrilaterals 7. Right Triangles 8. Circles 9. Surface Area & Volume 10. Coordinate Geometry

GEOMETRY SOL TEST BLUEPRINT (50 QUESTIONS TOTAL)

Resources

Text: Glencoe Geometry: Integration, Applications,

Connections, 1998, Glencoe McGraw-Hill HCPS Mathematics Website

http://blogs.henrico.k12.va.us/math/ HCPS Geometry Online

http://teachers.henrico.k12.va.us/math/HCPSgeo/

Reasoning, Lines, and Transformations 18 questions 36% of the Test

Triangles 14 questions 28% of the Test

Polygons, Circles, and Three-Dimensional Figures 18 questions 36% of the Test


http://blogs.henrico.k12.va.us/math/




Virginia Department of Education Mathematics SOL Resources http://www.doe.virginia.gov/testing/sol/standards_docs/mathematics/index.shtml

DOE Enhances Scope and Sequence Lesson Plans http://www.doe.virginia.gov/testing/sol/scope_sequence/mathematics_2009/

SOL Vertical Articulation Previous Standards Geometry Standard Future Standards

G.1 The student will construct and judge the validity of a logical

argument consisting of a set of premises and a conclusion. This will

include a) identifying the converse, inverse, and contrapositive of a

conditional statement; b) translating a short verbal argument into

symbolic form; c) using Venn diagrams to represent set relationships; &

d) using deductive reasoning.

Under Construction

A.6 graph linear equations/linear inequal (2 vars) ‐ a) determine slope of

line given equation of line/graph of line or two points on line ‐ slope as

rate of change; b) write equation of line given graph of line/two points

on line or slope‐point on line

8.6 a) verify/describe relationships among vertical/adjacent/

supplementary/complementary angles;

G.2 The student will use the relationships between angles formed by two

lines cut by a transversal to a) determine whether two lines are parallel;

b) verify the parallelism, using algebraic and coordinate methods as well

as deductive proofs; and c) solve real‐world problems involving angles

formed when parallel lines are cut by a transversal.

A.6 graph linear equations/linear inequal (2 vars) ‐ a) determine slope of

line given equation of line/graph of line or two points on line ‐ slope as

rate of change; b) write equation of line given graph of line/two points

on line or slope‐point on line

8.8 a) apply transformations to plane figures; b) ID applications of

transformations

7.8 represent transformations (reflections, dilations, rotations, and

translations) of polygons in the coordinate plane by graphing

G.3 The student will use pictorial representations, including computer

software, constructions, and coordinate methods, to solve problems

involving symmetry and transformation. This will include a)

investigating and using formulas for finding distance, midpoint, and

slope; b) applying slope to verify and determine whether lines are

parallel or perpendicular; c) investigating symmetry and determining

whether a figure is symmetric with respect to a line or a point; & d)

determining whether a figure has been translated, reflected, rotated, or

dilated, using coordinate methods.

6.12 determine congruence of segments/angles/polygons G.4 The student will construct and justify the constructions of a) a line

segment congruent to a given line segment; b) the perpendicular bisector

of a line segment; c) a perpendicular to a given line from a point not on

the line; d) a perpendicular to a given line at a given point on the line; e)

the bisector of a given angle, f) an angle congruent to a given angle; and

g) a line parallel to a given line through a point not on the given line.

G.5 The student, given information concerning the lengths of sides

and/or measures of angles in triangles, will a) order the sides by length,

given the angle measures; b) order the angles by degree measure, given

the side lengths; c) determine whether a triangle exists; & d) determine

the range in which the length of the third side must lie. These concepts

will be considered in the context of real‐world situations.

http://www.doe.virginia.gov/testing/sol/standards_docs/mathematics/index.shtml

http://www.doe.virginia.gov/testing/sol/scope_sequence/mathematics_2009/



7.6 determine similarity of plane figures and write proportions to express

relationships between similar quads and triangles

6.12 determine congruence of segments/angles/polygons

G.6 The student, given information in the form of a figure or statement,

will prove two triangles are congruent, using algebraic and coordinate

methods as well as deductive proofs.

G.7 The student, given information in the form of a figure or statement,

will prove two triangles are similar, using algebraic and coordinate


A.3 express sq roots/cube roots of whole numbers/the square root of

monomial alg exp (simplest radical form)

8.10 a) verify the Pythagorean Theorem; b) apply the Pythagorean

Theorem

G.8 The student will solve real‐world problems involving right triangles

by using the Pythagorean Theorem and its converse, properties of special

right triangles, and right triangle trigonometry.

A.4 solve multistep linear/ quad equation (2 vars) ‐ a) solve literal

equation; b) justify steps used in simplifying expresessions and solving

equations; c) solve quad equations (alg/graph); d) solve multistep linear

equations (alg/graph)

7.7 compare/contrast quadrilaterals based on properties

6.13 ID/describe properties of quadrilaterals

G.9 The student will verify characteristics of quadrilaterals and use

properties of quadrilaterals to solve real‐world problems.

A.4 solve multistep linear/ quad equation (2 vars) ‐ a) solve literal

equation; b) justify steps used in simplifying expresessions and solving

equations; c) solve quad equations (alg/graph); d) solve multistep linear

equations (alg/graph)

6.12 determine congruence of segments/angles/polygons

G.10 The student will solve real‐world problems involving angles of

polygons.

6.10 a) define π; b) solve practical problems w/circumference/area of

circle; c) solve practical problems involving area and perimeter given

radius/diameter; d) describe/determine volume/surface area of

rectangular prism

G.11 The student will use angles, arcs, chords, tangents, and secants to

a) investigate, verify, and apply properties of circles; b) solve real‐world

problems involving properties of circles; and c) find arc lengths and

areas of sectors in circles.

6.10 a) define π; b) solve practical problems w/ circumference/area of

circle; c) solve practical problems involving area and perimeter given

radius/diameter; d)describe/determine volume/surface area of

rectangular prism

G.12 The student, given the coordinates of the center of a circle and a

point on the circle, will write the equation of the circle.

8.7 a) investigate/solve practical problems involving volume/surface area

of prisms, cylinders, cones, pyramids; b) describe how changes in

measured attribute affects volume/surface area

8.9 construct a 3‐D model given top or bottom/side/front views

7.5 a) describe volume/surface area of cylinders; b) solve practical

problems involving volume/surface area of rect. prims and cylinders; c)

describe how changes in measured attribute affects volume/surface area

G.13 The student will use formulas for surface area and volume of three

dimensional objects to solve real‐world problems.

8.7 a) investigate/solve practical problems involving volume/surface area

of prisms, cylinders, cones, pyramids; b) describe how changes in

measured attribute affects volume/ surface area

7.5 a) describe volume/surface area of cylinders; b) solve practical

problems involving volume/surface area of rect. prims and cylinders; c)

describe how changes in measured attribute affects volume/surface area

G.14 The student will use similar geometric objects in two‐ or three

dimensions to a) compare ratios between side lengths, perimeters, areas,

and volumes; b) determine how changes in one or more dimensions of an

object affect area and/or volume of the object; c) determine how changes

in area and/or volume of an object affect one or more dimensions of the

object; and d) solve real‐world problems about similar geometric objects.



Fundamentals

Fir

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Peri

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Lessons

Point, Line, Plane

Segments and Rays

Angles

Pairs of Angles

Strand:

Grade 4, 5, 6: Geometry

SOL 4.10 The student will

a) identify and describe representations of

points, lines, line segments, rays, and

angles, including endpoints and vertices

5.12 The student will classify

a) angles as right, acute, obtuse, or

straight

6.12 The student will determine

congruence of segments, angles, and

polygons.

Return to Course Outline

Essential Knowledge and Skills

The student will use problem solving,

mathematical communication, mathematical

reasoning, connections, and representations to

Grade 4

Identify and describe representations of points,

lines, line segments, rays, and angles,

including endpoints and vertices.

Grade 5

Classify angles as right, acute, straight, or

obtuse.

Grade 6

Determine the congruence of segments,

angles, and polygons given their attributes.

Essential Understanding All students should

Grade 4

Understand that points, lines, line segments,

rays, and angles, including endpoints and

vertices are fundamental components of

noncircular geometric figures.

Understand that the shortest distance between

two points on a flat surface is a line segment.

Grade 5

Understand that angles can be classified as

right, acute, obtuse, or straight according

to their measures.

Grade 6

Given two congruent figures, what

inferences can be drawn about how the

figures are related? The congruent figures

will have exactly the same size and shape.

(continued)



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Fundamentals (continued) Resources

HCPS Geometry Online:

Unit 1 - Fundamental Concepts

Textbook: 1-2 Points, Lines, and Planes

1-4 Measuring Segments

1-6 Exploring Angles

1-7 Angle Relationships

Key Vocabulary

acute

angle

bisect

collinear

complementary

coplanar

horizontal

intersect

line

obtuse

plane

point

postulate

ray

right [angle]

segment

supplementary

theorem

vertex

vertical

Essential Questions

How are the concepts of points, lines, line segments, rays, angles,

endpoints, and vertices important when describing and comparing

geometric figures?

Where can we find points, lines, line segments, rays, and angles in

the world around us?

How can a set of intersecting lines be used to demonstrate the

relationship between and among points, lines, line segments, rays,

angles, and geometric figures?

How can visualizing a circle folded into halves and quarters help


us classify angles?

Given two congruent figures, what inferences can be drawn about

how the figures are related?

Teacher Notes and Elaborations

A point is a location in space. It has no length, width, or height. A

point is usually named with a capital letter.

A line is a collection of points going on and on infinitely in both

directions. It has no endpoints. When a line is drawn, at least two

points on it can be marked and given capital letter names. Arrows

must be drawn to show that the line goes on in both directions

infinitely (e.g., AB , read as “the line AB”).

A line segment is part of a line. It has two endpoints and includes all

the points between those endpoints. To name a line segment,

name the endpoints (e.g., AB , read as “the line segment AB”).

A ray is part of a line. It has one endpoint and continues infinitely in

one direction. To name a ray, say the name of its endpoint first

and then say the name of one other point on the ray (e.g., AB ,

read as “the ray AB”).

Two rays that have the same endpoint form an angle. This endpoint is

called the vertex. Angles are found wherever lines and line

segments intersect. An angle can be named in three different ways

by using

- three letters to name, in this order, a point on one ray, the

vertex, and a point on the other ray;

- one letter at the vertex; or

- a number written inside the rays of the angle.

Intersecting lines have one point in common. (continued)

http://teachers.henrico.k12.va.us/math/HCPSgeo/Fundamentals.html



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Fundamentals (continued) Teacher Notes and Elaborations (continued)

Congruent figures are figures having exactly the same size and shape.

Opportunities for exploring figures that are congruent and/or

noncongruent can best be accomplished by using physical

models.

A right angle measures exactly 90.

An acute angle measures greater than 0 but less than 90.

An obtuse angle measures greater than 90 but less than 180.

A straight angle forms an angle that measures exactly 180°.

Congruent figures have exactly the same size and the same shape.

Noncongruent figures may have the same shape but not the same size.

The symbol for congruency is .

The determination of the congruence or noncongruence of two

figures can be accomplished by placing one figure on top of the

other or by comparing the measurements of all sides and angles.

Construction of congruent line segments, angles, and polygons helps

students understand congruency.




Logic

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Lessons

Conditional Statements

Logic

Strand: Reasoning, Lines and

Transformations

SOL G.1 The student will construct and

judge the validity of a logical argument

consisting of a set of premises and a

conclusion. This will include

a) identifying the converse, inverse, and

contrapositive of a conditional

statement;

b) translating a short verbal argument

into symbolic form;

c) using Venn diagrams to represent set

relationships; and







Identify the converse, inverse, and

contrapositive of a conditional statement.

Translate verbal arguments into symbolic form

such as (p → q) and (~p → ~q).

Determine the validity of a logical argument.

Use valid forms of deductive reasoning,

including the law of syllogism, the law of the

contrapositive, the law of detachment, and

counterexamples.

Select and use various types of reasoning and

methods of proof, as appropriate.

Use Venn diagrams to represent set

relationships, such as intersection, and union.

Interpret Venn diagrams.

Recognize and use the symbols of formal logic,

which include , , ~, , ᴧ and ᴠ.

Essential Understanding Inductive reasoning, deductive reasoning, and

proof are critical in establishing general claims.

Deductive reasoning is the method that uses

logic to draw conclusions based on definitions,

postulates, and theorems.

Inductive reasoning is the method of drawing

conclusions from a limited set of observations.

Logical arguments consist of a set of premises

or hypotheses and a conclusion.

Proof is a justification that is logically valid

and based on initial assumptions, definitions,

postulates, and theorems.

Euclidean geometry is an axiomatic system

based on undefined terms (point, line, and

plane), postulates, and theorems.

When a conditional and its converse are true,

the statements can be written as a biconditional

(i.e., iff or if and only if).

Logical arguments that are valid may not be

true. Truth and validity are not synonymous.

(continued)



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Logic (continued) Resources


Reasoning

Textbook: 2-2 If-Then Statements and Postulates

2-3 Deductive Reasoning

DOE ESS Lesson Plans:

Lines and Angles (PDF) (Word)

Key Vocabulary biconditional statement

conclusion

conditional statement

conjecture

contrapositive

converse

counterexample

deductive reasoning

hypothesis (premise)

inductive reasoning

inverse

Law of Detachment

Law of Syllogism

Law of the

Contrapositive

postulate (axiom)

proof

symbolic form

Venn diagram


Essential Questions

When is a statement a lie?

What is the importance or need for symbolic representation of

words?

What does it mean to be logical?

How can logic be represented visually?

What is the relationship between reasoning, justification, and

proof in geometry?

What is a truth-value?

How does a truth-value apply to conditional statements?

How do deductive reasoning and Venn diagrams help judge the

validity of logical arguments?


Logic is the study of the principles of reasoning. Logical arguments

consist of a set of premises (hypotheses) and a conclusion (the last

step in a reasoning process).

Terms associated with logical arguments are reasoning, justification,

and proof. Reasoning is the drawing of conclusions or inferences from

facts, observations, or hypotheses. Justification is a rationale or

argument for some mathematical proposition. A conjecture is a

statement that has not been proved true nor shown to be false. A proof

is a justification that is logically valid and based on initial

assumptions, definitions, and proven results. A theorem is a statement

that can be proved and a postulate or axiom is an assumption (a

statement taken for granted) that is accepted without proof. A

justification may be less formal than a proof. It may consist of a set of

examples that seem to support the proposition or it may be an

intuitive argument. The three concepts are related in that reasoning is

used to seek a justification of a proposition, which, if possible, is

turned into a proof. (continued)

http://teachers.henrico.k12.va.us/math/HCPSgeo/G-1.html

http://www.doe.virginia.gov/testing/solsearch/sol/math/G/m_ess_g-2.pdf

http://www.doe.virginia.gov/testing/solsearch/sol/math/G/m_ess_g-2.doc



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Logic (continued) Teacher Notes and Elaborations (continued)

Communication of reasoning and/or justification to complete a proof

can be shown through symbolic form (truth tables or Venn diagrams)

or written form (paragraph, indirect, two-column or coordinate

method).

An if-then statement is called a conditional statement or simply a

conditional. A conditional statement includes an initial condition or

hypothesis (premise) and its corresponding outcome (conclusion). The

conditional statement is written in the if (hypothesis) – then

(conclusion) form.

q

If p (hypothesis), then q (conclusion). p

The converse (a proposition produced by reversing position or order)

of the conditional statement is formed by interchanging the hypothesis

and its conclusion.

p

If q (conclusion), then p (hypothesis).

q

The inverse of the conditional statement is formed by negating both

the hypothesis and the conclusion.

If not p (hypothesis), then not q (conclusion).

The contrapositive of the conditional statement is formed by

interchanging and negating both the hypothesis and the conclusion.

If not q (conclusion), then not p (hypothesis).

The contrapositive and original conditional statements are logically

equivalent (Law of Contrapositive).


Symbolic form includes truth tables (tabular representation of the

truth or falsehood of hypotheses and conclusions) and Venn diagrams.

Deductive reasoning uses rules to make conclusions. Applying the

Law of Detachment, if you accept “If p then q” as true and you accept

p as true, then you must logically accept q as true. It also follows if

you accept “If p then q” as true and you accept not q as true, then you

must logically accept not p as true. According to the Law of

Syllogism, if you accept “If p then q” as true and if you accept “If q

then r” as true, then you must logically accept “If p then r” as true. A

counterexample is an example used to prove an if-then statement

false. For that counterexample, the hypothesis is true and the

conclusion is false.

Inductive reasoning is a kind of reasoning in which the conclusion is

based on several past observations.

Symbolically means “therefore”. Ex: m ABC is 90° m ABCis a right angle.

In logic, letters are used to represent simple statements that are either

true or false. Simple statements can be joined to form compound

statements. A conjunction is a compound statement composed of two

simple statements joined by the word “and”. The symbol , is used to

represent the word “and”. A disjunction is a compound statement x

(continued)



Logic (continued)

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Extension for PreAP Geometry

Identify, create, and determine the truth-value of the converse,

inverse, and contrapositive of a conditional statement.

Use chain reasoning to make a logical conclusion given a set of

statements.

Identify logically equivalent statements.

Construct truth tables given statements (conditional, conjunction,

disjunction, biconditionals, etc.).

Investigate the concept of an indirect proof.


The truth value of a statement is either true or false. A truth table can

be used to determine the conditions under which a statement is true.

Truth Tables:

Conditional

If p then q

p q p q

T T T

T F F

F T T

F F T

Conjunction

p and q

p q p q

T T T

T F F

F T F

F F F


Disjunction

p or q p q p q

T T T

T F T

F T T

F F F

An indirect proof is a proof that begins by assuming temporarily that

the conclusion is not true; then reason logically until a contradiction

of the hypothesis or another known fact is reached.

Sample Instructional Strategies and Activities

Students, working in cooperative learning groups, will solve logic

problems to introduce the concept of deductive reasoning. Each

group of students will give their solutions and describe their

thought processes.



Lines

Fir

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Pairs of Lines

Angles and Parallel Lines

Proving Lines Parallel

Strand: Reasoning, Lines and

Transformations

SOL G.2 The student will use the

relationships between angles formed by

two lines cut by a transversal to

a) determine whether two lines are

parallel;

b) verify the parallelism, using algebraic

and coordinate methods as well as

deductive proofs; and

c) solve real-world problems involving

angles formed when parallel lines are

cut by a transversal.






Use algebraic and coordinate methods as well

as deductive proofs to verify whether two lines

are parallel.

Solve problems by using the relationships

between pairs of angles formed by the

intersection of two parallel lines and a

transversal including corresponding angles,

alternate interior angles, alternate exterior

angles, and same-side (consecutive) interior

angles.

Solve real-world problems involving

intersecting and parallel lines in a plane.

Essential Understanding Euclidean geometry is an axiomatic system

based on undefined terms (point, line, and

plane), postulates, and theorems.

When a conditional and its converse are true,

the statements can be written as a biconditional

(i.e., iff or if and only if).

Logical arguments that are valid may not be

true. Truth and validity are not synonymous.

Parallel lines intersected by a transversal form

angles with specific relationships.

Some angle relationships may be used when

proving two lines intersected by a transversal

are parallel.

The Parallel Postulate differentiates Euclidean

from non-Euclidean geometries such as

spherical geometry and hyperbolic geometry.

(continued)



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Lines (continued) Resources


Lines

Textbook: 3-1 Parallel Lines and Transversals

3-2 Angles and Parallel Lines

3-3 Slopes of Lines

3-4 Proving Lines Parallel


Lines and Angles (PDF) (Word)

Key Vocabulary adjacent angles

algebraic method

alternate exterior

angles

alternate interior angles

complementary angles

consecutive (same-

side) interior angles

coordinate method

corresponding angles

deductive reasoning

inductive reasoning

exterior angle

interior angle

linear pair

parallel

perpendicular

skew

supplementary angles

transversal

union

vertical angles


Essential Questions

What is the relationship between lines and angles?

What is the difference between parallel lines and perpendicular

lines?

How are lines proven parallel?

What is the difference between parallel lines and intersecting

lines?

What are the relationships between the angles formed when two

parallel lines are cut by a transversal?

Teacher Notes and Elaborations Euclidean Geometry is a

mathematical system attributed to the Alexandrian Greek

mathematician Euclid, whose elements is the earliest known

systematic discussion of geometry. Euclid's method consists in

assuming a small set of intuitively appealing axioms, and deducing

many other theorems (propositions) from these.

Angles with the same measure are congruent angles. Adjacent angles

are two angles that share a common side and have the same vertex,

but have no interior points in common. Vertical angles are two angles

whose sides form two pairs of opposite rays. When two lines

intersect, they form two pairs of vertical angles.

When two lines intersect, two types of angle pairs are formed: vertical

angles and adjacent supplementary angles. Vertical angles are

congruent and two adjacent angles are supplementary.

Parallel lines are lines that are in the same plane (coplanar) and never

intersect because they are always the same distance apart. They have

no points in common. The symbol || indicates parallel lines. Skew

lines do not intersect and are not coplanar.

(continued)






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Lines (continued) Teacher Notes and Elaborations G.2

Intersection is a point or set of points common to two or more figures.

A transversal is a line that intersects two or more coplanar lines in

different points forming eight angles. Interior angles lie between the

two lines. Alternate interior angles are on opposite sides of the

transversal. Consecutive interior angles are on the same side of the

transversal. Exterior angles lie outside the two lines. Alternate

exterior angles are on opposite sides of the transversal.

Corresponding angles are nonadjacent angles located on the same

side of the transversal where one angle is an interior angle and the

other is an exterior angle.

If the sum of the measures of two angles is 180°, then the two angles

are supplementary. If the two angles are adjacent and supplementary

then they are a linear pair.

If the sum of the measures of two angles is 90°, then the two angles

are complementary. If the two angles are adjacent and complementary

then they form a right angle.

If two lines in a plane are cut by a transversal, the lines are parallel if:

- alternate interior angles are congruent,

- alternate exterior angles are congruent,

- corresponding angles are congruent,

- same side (consecutive) interior angles are supplementary.

Proving lines parallel implies determining whether necessary and

sufficient conditions (properties, definitions, postulates, and

theorems) exist for parallelism. A proof is a chain of logical

statements starting with given information and leading to a

conclusion.


Two column deductive proofs (formal proofs) are examples of

deductive reasoning. They contain statements and reasons organized

in two columns. Each step is called a statement, and the properties

that justify each step are called reasons.

Essential parts of a good proof include:

1. state the theorem or conjecture to be proven;

2. list the given information;

3. if possible, draw a diagram to illustrate the given

information;

4. state what is to be proved; and

5. develop a system of deductive reasoning.

The Parallel Postulate is the axiom of Euclidean Geometry stating

that if two straight lines are cut by a third, the two will meet on the

side of the third on which the sum of the interior angles is less than

two right angles. Equivalently, Playfair’s Axiom states: “If given a

line and a point not on the line, then there exists exactly one line

through the point that is parallel to the given line.” In Euclidean

Geometry, parallel lines lie in the same plane and never intersect. In

spherical geometry, the sphere is the plane, and a great circle

represents a line.

Two nonvertical coplanar lines are parallel if and only if their slopes

are equal. Two nonvertical coplanar lines are perpendicular if and

only if the product of their slopes is 1 .

Algebraic and coordinate methods should also be used to determine

parallelism. Coordinate geometry establishes a correspondence

between algebraic concepts and geometric concepts. For example, the

distance formula is derived as an application of the Pythagorean

Theorem. The Pythagorean Theorem in turn is used to develop the

equation of a circle. The coordinate proof is often more convenient



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Lines (continued) Teacher Notes and Elaborations G.2 (continued)

than a two-column proof. The following is an example of a

coordinate proof involving parallelism.

Prove: The segment that joins the midpoint of two sides of a

triangle is parallel to the third side.

Given: OAB and M and N the midpoints of OB and OA

respectively.

Prove: MN || BA

Proof: Choose axes and coordinates as shown.

y

B (2 ,2 )b c

M

O N A (2 ,0)a x

1. Midpoints are 2 0 2 0 2 2

M( , ) ( , ) ( , )2 2 2 2

b c b cb c

and

2 0 0 0 2 0N( , ) ( , ) ( ,0)

2 2 2 2

a aa

; by Midpoint Formula.

2. Slope of 0

MNc c

a b a b

and the slope of

0 2 2BA

2 2 2( )

c c c

a b a b a b

; by definition of slope.

3. Slope of MN = slope of BA ; by Substitution Property. MN || BA ;

two nonvertical lines are parallel if and only if their slopes are

equal.



Skew lines are non-coplanar lines that do not intersect. Experiences

with skew lines should include 3-dimensional models.

Extension for PreAP Geometry Use algebraic, coordinate, and deductive methods to determine if

lines are perpendicular.

Write equations of parallel and perpendicular lines.

Investigate skew lines using real world models.

Use definitions, postulates, and theorems to complete two-column

or paragraph proofs with at least five steps.

Extension for PreAP Geometry In a paragraph proof (informal proof) a paragraph is written to explain

why a conjecture for a given situation is true.


Have students pick two lines on notebook paper. Use straight edge

and pencil to darken lines chosen. Using a straight edge, draw a

transversal. Label angles. Have students accurately measure pairs

of special angles using a protractor. Perform the same procedures

with two non-parallel lines cut by a transversal. Write conjectures

for each special angle pair (corresponding, consecutive interior,

alternate interior, and alternate exterior).

Use patty paper to trace and compare lines and angles.

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Lines (continued) Sample Instructional Strategies and Activities

Have class look for parallel, intersecting, perpendicular, and skew

lines in the classroom. In groups, students list as many pairs of

them as they can find in ten minutes. Each group gives some

examples from their list. This can be used as a competition.

o Have students pick two lines on notebook paper. Use

straight edge and pencil to darken lines chosen. Using a

straight edge, sketch a transversal. Label angles. Have

students accurately measure pairs of special angles. Use

the same procedure with two non-parallel lines cut by a

transversal. Write conjectures for each special angle pair

(corresponding, consecutive interior, alternate interior, and

alternate exterior).

o Take class outside to look for parallel, intersecting,

perpendicular, and skew lines and for identified angles. In

groups, students list as many pairs of them as they can find

in ten minutes. After returning to the classroom, each

group gives some examples from their list. This can be

used as a competition.

Have students use patty paper to discover congruent angles

formed when parallel lines are cut by a transversal.

Have students build an angle log book. Students will draw

pictures of various angles and label the angle. Students will relate

the angle to an object in the room.




Constructions

Fir

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Lessons

Constructions

Strand: Reasoning, Lines, and

Transformations

SOL G.4 The student will construct and justify

the constructions of

a) a line segment congruent to a

given line segment;

b) the perpendicular bisector of a line

segment;

c) a perpendicular to a given line

from a point not on the line;

d) a perpendicular to a given line at a

given point on the line;

e) the bisector of a given angle;

f) an angle congruent to a given

angle; and

g) a line parallel to a given line

through a point not on the given

line.






Construct and justify the constructions of

- a line segment congruent to a given line

segment;

- the perpendicular bisector of a line

segment;

- a perpendicular to a given line from a

point not on the line;

- a perpendicular to a given line at a point

on the line;

- the bisector of a given angle;

- an angle congruent to a given angle; and

- a line parallel to a given line through a

point not on the given line.

Construct an equilateral triangle, a square, and

a regular hexagon inscribed in a circle.

Construct the inscribed and circumscribed

circles of a triangle.

Construct a tangent line from a point outside a

given circle to the circle.

Essential Understanding

Construction techniques are used to solve real-

world problems in engineering, architectural

design, and building construction.

Construction techniques include using a

straightedge and compass, paper folding, and

dynamic geometry software.

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Constructions (continued) Resources


Constructions

Textbook: pg. 31 A line segment congruent to a given line segment

pg. 39 The perpendicular bisector of a line segment

pg. 56 A perpendicular to a given line from a point not on the line

pg. 56 A perpendicular to a given line at a point on the line

pg. 48 The bisector of a given angle

pg. 47 An angle congruent to a given angle

pg. 146 A line parallel to a given line through a point not on the

given line


Constructions (PDF) (Word)

Key Vocabulary

bisector

centroid

circumcenter

circumscribed

compass

congruence

construction

incenter

inscribed

intersection

parallel lines

perpendicular bisector

perpendicular lines

bisector

straightedge

transversal

Essential Questions

What is the relationship between points, rays, and angles?

Why are constructions important?

How are constructions justified?



"Construction" in geometry means to draw shapes, angles or lines

accurately. Constructions are done using tools including software

programs such as Sketchpad, Geogebra, patty paper, a straightedge,

and a compass. If students are using a ruler as a straightedge, they

should be instructed to ignore its markings. Constructions help build

an understanding of the relationships between lines and angles. The

seven basic constructions can be used to do more complicated

constructions such as points of concurrency: centroid, incenter,

circumcenter and orthocenter.

The intersection of two figures is the set of points that is in both

figures.

A transversal is a line that intersects two or more coplanar lines in

different points.

Two angles are congruent if and only if they have equal measures. A

ray is an angle bisector if and only if it divides the angle into two

congruent adjacent angles.

Parallel lines are lines that do not intersect and are coplanar.

Perpendicular lines are lines that intersect at right angles. A segment

bisector is a line, segment, ray, or plane that intersects the segment at

its midpoint. A perpendicular bisector of a segment is a line, ray, or

segment that is perpendicular to the segment at its midpoint.

A circle is circumscribed about a triangle if the circle contains all the

vertices of the triangle. A triangle is inscribed in a circle if each of its

vertices lies on the circle.

In a triangle, a median is a segment that joins a vertex of the triangle (continued)






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Constructions (continued) Teacher Notes and Elaborations (continued) and the midpoint of the side opposite that vertex. The medians of a

triangle intersect at the balance point called the centroid.

To circumscribe a circle about a triangle, construct the perpendicular

bisectors of each side. The point where these perpendicular bisectors

meet is the circumcenter. Using the circumcenter and any vertex of

the triangle as the radius, construct the circle about the triangle.

To construct a circle inscribed inside a triangle, construct the angle

bisectors. The incenter is the point where the angle bisectors meet.

Construct a perpendicular from the incenter to one of the sides of the

triangle. This perpendicular segment is the radius of the inscribed

circle.

Justification of constructions may involve application of postulates,

theorems, definitions, and properties. Justification of constructions

may differ depending upon the plan proposed, and the order in which

concepts are taught.

Construction Justification 1. Construct a line segment Radii of equal circles are equal

congruent to a given a line

segment

2. Construct an angle congruent Radii of equal circles are equal

to a given angle SSS Postulate

Corresponding parts of

congruent triangles are

congruent

3. Construct the bisector of a Radii of equal circles are equal

given angle SSS Postulate

Corresponding parts of

congruent triangles are

congruent

Definition of an angle bisector Return to Course Outline

4. Construct the perpendicular Radii of equal circles are equal

bisector of a given segment Through any two points there is

exactly one line

If a point is equidistant from the

endpoints of a line segment, then

the point lies on the perpendic-

ular bisector of the line segment

5. Construct the perpendicular Radii of equal circles are equal

to a line at the given point on Definition of a straight angle

the line. Definition of an angle bisector

Definition of right angles and

definition of perpendicular

lines

6. Construct the perpendicular to Radii of equal circles are equal

the line from a point not on the If a point is equidistant from the

line. endpoints of a line segment,

then the point lies on the

perpendicular bisector of the

line

7. Construct the parallel to a Radii of equal circles are equal

given line though a given If two lines are cut by a

point not on the line. transversal and corresponding

angles are congruent, then the

lines are parallel


Construct angles with measures of 15, 30, 45, 60, 75, and 135

degrees.

Construct a tangent to a circle through a point on the circle.

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Constructions (continued) Extension for PreAP Geometry (continued)

Justify the constructions of:

- equilateral triangles;

- squares;

- angles with measures of 15, 30, 45, 60, 75, and 135 degrees;

- regular hexagons;

- a tangent to a circle through a point on the circle; and

- a tangent line from a point outside a given circle to the circle.

Given a segment, by construction divide the segment into a given

number of congruent parts.




Triangle Fundamentals

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Lessons

Triangle Fundamentals

Isosceles Triangles

Triangle Inequalities

Strand: Triangles; Polygons and

Circles

SOL G.5

The student, given information

concerning the lengths of sides and/or

measures of angles in triangles, will

a) order the sides by length, given the

angle measures;

b) order the angles by degree

measure, given the side lengths;

c) determine whether a triangle

exists; and

d) determine the range in which the

length of the third side must lie.

These concepts will be considered in

the context of real-world situations.

SOL G.10 The student will solve real-world

problems involving angles of

polygons.






Order the sides of a triangle by their lengths

when given the measures of the angles.

Order the angles of a triangle by their

measures when given the lengths of the sides.

Given the lengths of three segments, determine

whether a triangle could be formed.

Given the lengths of two sides of a triangle,

determine the range in which the length of the

third side must lie.

Solve real-world problems given information

about the lengths of sides and/or measures of

angles in triangles.

Solve real-world problems involving the

measures of interior and exterior angles of

polygons.

Identify tessellations in art, construction, and

nature.

Find the sum of the measures of the interior

and exterior angles of a convex polygon.

Find the measure of each interior and exterior

angle of a regular polygon.

Find the number of sides of a regular polygon,

given the measures of interior or exterior

angles of the polygon.


The longest side of a triangle is opposite the

largest angle of the triangle and the shortest

side is opposite the smallest angle.

In a triangle, the length of two sides and the

included angle determine the length of the

side opposite the angle.

In order for a triangle to exist, the length of

each side must be within a range that is

determined by the lengths of the other two

sides.

Two intersecting lines form angles with

specific relationships.

The exterior angle and the corresponding

interior angle form a linear pair.

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Triangle Fundamentals (continued) Resources


Triangle Inequalities

Textbook: 4-1 Classifying Triangles

4-2 Measuring Angles in Triangles

4-6 Analyzing Isosceles Triangles

5-4 Inequalities for Sides and Angles of a Triangle

10-1 Polygons

10-2 Tessellations


How Many Triangles? (PDF) (Word)

Angles in Polygons (PDF) (Word)

Key Vocabulary

altitude

apothem

concave

convex

decagon

diagonal

dodecagon

exterior angle

heptagon

hexagon

interior angle

isosceles

linear pair

median

n-gon

nonagon

octagon

opposite

pentagon

polygon

quadrilateral

regular/irregular polygon

scalene

tessellation

tiling

triangle

Triangle Inequality Theorem


Essential Questions

What are the angle relationships of a triangle?

What conditions must exist for a triangle to be formed?

What is the relationship between the measure of the angles and the

lengths of the opposite sides of a triangle?


Triangle Inequality Theorem: The sum of the lengths of any two sides

of a triangle is greater than the length of the third side.

If one side of a triangle is longer than another side, then the angle

opposite (across from) the longer side is larger than the angle opposite

the shorter side.

If one angle of a triangle is larger than another angle, then the side

opposite the larger angle is longer than the side opposite the smaller

angle.

Sides of a triangle can be put in order when given the measures of the

angles. If the sides of a triangle are ordered longest to shortest then the

angles opposite must also be ordered largest to smallest.


Use the Hinge Theorem and its converse to compare side lengths

and angle measures in two triangles.

Given a quadrilateral with one diagonal, write inequalities relating

pairs of angles or segment measures.

(continued)








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Triangle Fundamentals (continued) Extension for PreAP Geometry

Using properties of triangles, inequalities can be written relating pairs

of angles or segment measures. Note: Figures are not drawn to scale

BCD CAB

CD BC

Hinge Theorem: (SAS Inequality) If two sides of a triangle are

congruent to two sides of another triangle, and the included angle in

one triangle is greater than the included angles in the other, then the

third side of the first triangle is longer than the third side in the second

triangle.



Coordinate geometry can be used to investigate relationships

among triangles.

Use pieces of yarn, straws, sticks, or magnetic tape to see which

combinations of lengths can be used to make triangles.

Use Geo-Legs or Anglegs to illustrate combinations of lengths that

can be used to form triangles.

Cut out a triangle. Place a different color dot in each angle. Place

the triangle on the paper and trace around it in pencil. Slide triangle

over and mark the color in each angle so that the colors correspond

with the cardboard triangle. Place triangle back on top and rotate it

so that it no longer overlaps. Repeat until the plane is filled. Have

students identify parallel lines, vertical angles, etc. Students make

conjectures about lines and angles in the tessellation. Students are

given various polygons and asked if they tessellate a plane. Explain

why or why not.

\



Proofs with Congruent Triangles

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Lessons

Congruent Triangles

Proving Triangles Congruent –

SSS, SAS, ASA

Proving Triangles Congruent –

AAS, HL

Strand: Triangles

SOL G.6 The student, given information in the

form of a figure or statement, will

prove two triangles are congruent,

using algebraic and coordinate







Use definitions, postulates, and theorems to

prove triangles congruent.

Use coordinate methods, such as the distance

formula and the slope formula, to prove two

triangles are congruent.

Use algebraic methods to prove two triangles

are congruent.


Congruence has real-world applications in a

variety of areas, including art, architecture,

and the sciences.

Congruence does not depend on the position

of the triangle.

Concepts of logic can demonstrate

congruence or similarity.

Congruent figures are also similar, but similar

figures are not necessarily congruent.

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Proofs with Congruent Triangles (continued) Resources


Proofs

Textbook: 4-3 Exploring Congruent Triangles

4-4 Proving Triangles Congruent

4-5 More Congruent Triangles

5-2 Right Triangles


Congruent Triangles (PDF) (Word)

Key Vocabulary

AAS Theorem

algebraic methods

altitude

ASA Postulate

coordinate methods

corresponding parts

deductive proof

definition

distance formula

HL Postulate

hypotenuse

included angle

included side

leg

postulate

properties

SAS Postulate

SSS Postulate

theorem

Essential Questions

What are congruent triangles?

What are the one-to-one correspondences that prove triangles

congruent?

How can congruent triangles assist in the proof of other geometric

ideas?


When two figures have exactly the same shape and size, they are said

to be congruent. Using algebraic methods, if all corresponding parts


can be shown to be equal, then the figures are congruent. This can

include coordinate methods such as distance formula and the slope

formula.

Congruent figures have corresponding parts (matching parts) that

have equal measures. Corresponding parts of congruent triangles are

congruent (CPCTC).

Congruence does not depend on the position of the triangle.

A theorem is a statement that can be proved and a postulate is an

assumption that is accepted without proof. Definitions, postulates, and

theorems are used in proofs. A proof is a chain of logical statements

starting with given information and leading to a conclusion. Two

column deductive proofs are examples of deductive reasoning.

Properties (facts about real numbers and equality from algebra) can

also be used to justify steps in proofs.

A side of a triangle is said to be included (included side) between two

angles if the vertices of the two angles are the endpoints of the side.

An angle of a triangle is said to be included (included angle) between

two sides if the angle is formed by the two sides.

Triangles can be proven congruent with the following

correspondences:

SSS Postulate: Three sides of one triangle are congruent to the

corresponding sides of another triangle.

SAS Postulate: Two sides and the included angle of one triangle are

congruent to the corresponding two sides and included angle of

another triangle.

ASA Postulate: Two angles and the included side of one triangle

are congruent to the corresponding two angles and included side of

another triangle. (continued)






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Proofs with Congruent Triangles (continued) Teacher Notes and Elaborations (continued)

AAS Theorem: Two angles and a non-included side of one triangle

are congruent to the corresponding two angles and a non-included

side of a second triangle.

In a right triangle the side opposite the right angle is the hypotenuse

and the other two sides are called legs.

Right triangles can be proven congruent with the following

correspondence:

HL Postulate: The hypotenuse and a leg of one right triangle are

congruent to the hypotenuse and leg of another right triangle.

Medians, altitudes, and perpendicular bisectors are also used in

proving triangles congruent. A median of a triangle is a segment that

joins a vertex to the midpoint of the opposite side. An altitude of a

triangle is a segment from a vertex and perpendicular segment from a

vertex to the line containing the opposite side.


Use angle bisectors, medians, altitudes, perpendicular bisectors to

prove triangles congruent.

Correlate LL, HA, LA to SAS, AAS, and ASA respectively.

Investigate the points of concurrency of the lines associated with

triangles (angle bisectors (incenter), perpendicular bisectors

(circumcenter), altitudes (orthocenter), and medians (centroid)).


LL Theorem: The legs of one right triangle are congruent to the legs

of another right triangle.

HA Theorem: The hypotenuse and an acute angle of one right

triangle are congruent to the hypotenuse and acute angle of the other Return to Course Outline

right triangle.

LA Theorem: One leg and an acute angle of one right triangle are

congruent to the corresponding parts of another right triangle.

The medians of a triangle intersect at the common point called the

centroid.

In a triangle, the point where the perpendicular bisectors of each side

intersect is the circumcenter.

In a triangle, the incenter is the point where the angle bisectors

intersect.

In a triangle, the orthocenter is the point of intersection of the three

altitudes.

Sample Instructional Strategies and Activities Use coordinate geometry to investigate relationships among

triangles.

Given specifications such as side lengths or angle measures,

students draw a triangle. Next, the students compare their drawings

to see if they are congruent. This is done to test AAS, SSS, etc.

before they are introduced.

Students are given a printed deductive proof of theorem. Cut it up

into a statement of theorem, given, prove, diagram, individual

statements, and individual reasons. Each group of students is given

a set of pieces and must put the proof together in correct order.

Use pieces of yarn, straws, or sticks to see which combinations of

lengths can be used to make triangles.

Use patty paper to demonstrate congruent triangles.



Similarity

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Lessons

Using Proportions

Exploring Similar Polygons

Identifying Similar Triangles

Strand: Triangles

SOL G.7 The student, given information in the

form of a figure or statement, will

prove two triangles are similar, using

algebraic and coordinate methods as

well as deductive proofs.







prove triangles similar.

Use algebraic methods to prove that triangles

are similar.


formula, to prove two triangles are similar.

Compare ratios between side lengths,

perimeters, areas, and volumes, given two

similar figures.

Solve real-world problems involving measured

attributes of similar objects.


Similarity has real-world applications in a


and the sciences.

Similarity does not depend on the position of

the triangle.



A constant ratio exists between corresponding

lengths of sides of similar figures.

Proportional reasoning is integral to

comparing attribute measures in similar

objects

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Similarity (continued) Resources


Similarity

Textbook: 7-1 Using Proportions

7-2 Exploring Similar Polygons

7-3 Identifying Similar Triangles

7-4 Parallel Lines and Proportional Parts

7-5 Parts of Similar Triangles


Similar Triangles (PDF) (Word)

Key Vocabulary

area

deductive proof

perimeter

proportion

ratio

scale factor

similar figures

similar triangles

AA Similarity

SSS Similarity

SAS Similarity

volume

Essential Questions

When is a proportion necessary to solve a problem?

Are common units of measure necessary when solving

proportions?

How are similar triangles utilized in art, architecture and the

sciences?

What is the difference between congruence and similarity?

What is the relationship between similar triangles and proportions?

What are the one-to-one correspondences that prove triangles

similar? Return to Course Outline

In similar figures, how does a change of one measurement affect

perimeter, area, or volume?

Teacher Notes and Elaborations Congruent figures have corresponding parts that have equal measures

while similar figures have corresponding angles congruent but

corresponding sides with proportional measures. Coordinate methods

such as distance formula and the slope formula can be used to prove

triangles are similar.

A theorem is a statement that can be proved and a postulate is an

assumption that is accepted without proof. Definitions, postulates, and

theorems are used in proofs. A proof is a chain of logical statements

starting with given information and leading to a conclusion. Two

column deductive proofs are examples of deductive reasoning.

Properties (facts about real numbers and equality from algebra) can

also be used to justify steps in proofs.

A ratio is a comparison of two quantities. The ratio of a to b can be

expressed as a

b, where b 0. If two ratios are equal, then a proportion

exists. Therefore a c

b d is a proportion and the cross products are

equal (ad = bc).

Similar figures are figures that have the same shape but not

necessarily the same size. Two triangles are similar if and only if their

corresponding angles are congruent and the measures of their

corresponding sides are proportional. The ratio of the lengths of two

corresponding sides of two similar polygons is called a scale factor.

An angle of a triangle is said to be included (included angle) between

two sides if the angle is formed by the two sides. (continued)






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Similarity (continued) Teacher Notes and Elaborations (continued) There are three ways to determine whether two triangles are similar

when all measurements of both triangles are not known:

AA Similarity: Show that two angles of one triangle are

congruent to two angles of the other.

SSS Similarity: Show that the measures of the corresponding

sides of the triangles are proportional.

SAS Similarity: Show that the measures of two sides of a triangle

are proportional to the measures of the corresponding sides of the

other triangle and that the included angles are congruent.

If a line is drawn parallel to one side of a triangle and intersects the

other two sides, then it separates the sides into segments of

proportional lengths.

a c a c

b d

b d

If two triangles are similar, then the measures of the lengths of the

corresponding angle bisectors of the triangles are proportional to the

measures of the lengths of the corresponding sides.

a ~ c

x y

x a

y c

A median of a triangle is a segment that joins a vertex to the midpoint

of the opposite side. Return to Course Outline

If two triangles are similar, then the measures of the corresponding

medians are proportional to the measures of the corresponding sides.

~ c

a x y

x a

y c

An angle bisector in a triangle separates the opposite side into

segments that have the same ratio as the other two sides.

a b

e a

f b

e f


Use definitions, postulates, and theorems to complete two-column

or paragraph proofs with at least five steps.

Investigate proportionality in a triangle intersected by three or more

parallel lines.

Investigate the Golden Ratio.


If three or more parallel lines intersect two transversals, then they cut

off the transversals proportionally.

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Similarity (continued) Extension for PreAP Geometry (continued)

A B C

D

E

F

AB DE

=BC EF

, AC BC

=DF EF

, AC DF

=BC EF

If a line segment is divided into two lengths such that the ratio of the

segments’ entire length to the longer length is equal to the ratio of the

longer length to the shorter length, then the segment has been divided

into the Golden Ratio.

a b

a b a

a b

(This golden ratio is approximately 1.618.)

In a rectangle, if the ratio of the longer side to the shorter

approximates 1.618, the rectangle is called a Golden Rectangle


Use coordinate geometry to investigate relationships among

triangles.

Students are given a printed deductive proof of theorem. Cut it up

into a statement of theorem, given, prove, diagram, individual Return to Course Outline

statements, and individual reasons. Each group of students is given

a set of pieces and must put the proof together in correct order.

Each group of students will measure the height of one of their

members, the shadow of that member, and the shadow of a light

pole or flagpole. Using similar triangles and proportions, each

group calculates the height of the pole. Next, the groups compare

their calculations.

Given the pitch of a roof, the students will calculate the roof truss

and using toothpicks will construct a model of the roof.

Use patty paper to demonstrate similar triangles.

Using cylinders made from PVC pipe or empty cans determine the

change in volume with respect to changes in height or radius. Fill

cylinders with water to compare the volumes.



Right Triangles

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Lessons

Geometric Mean (optional)

The Pythagorean Theorem

Special Right Triangles

Trigonometry

Strand: Triangles

SOL G.8

The student will solve real-world

problems involving right triangles by

using the Pythagorean Theorem and

its converse, properties of special right

triangles, and right triangle

trigonometry.






Determine whether a triangle formed with

three given lengths is a right triangle.

Solve for missing lengths in geometric figures,

using properties of 45-45-90 triangles.

Solve for missing lengths in geometric figures,

using properties of 30-60-90 triangles.

Solve problems involving right triangles, using

sine, cosine, and tangent ratios.

Solve real-world problems, using right triangle

trigonometry and properties of right triangles.

Explain and use the relationship between the

sine and cosine of complementary angles.


prove triangles similar.

Use algebraic methods to prove that triangles

are similar.


formula, to prove two triangles are similar.


The Pythagorean Theorem is essential for

solving problems involving right triangles.

Many historical and algebraic proofs of the

Pythagorean Theorem exist.

The relationships between the sides and

angles of right triangles are useful in many

applied fields.

Some practical problems can be solved by

choosing an efficient representation of the

problem.

Another formula for the area of a triangle is

1sin

2A ab C .

The ratios of side lengths in similar right

triangles (adjacent/hypotenuse or

opposite/hypotenuse) are independent of the

scale factor and depend only on the angle the

hypotenuse makes with the adjacent side, thus

justifying the definition and calculation of

trigonometric functions using the ratios of

side lengths for similar right triangles.

Similarity has real-world applications in a


and the sciences.

Similarity does not depend on the position of

the triangle.



(continued)



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Right Triangles (continued) Resources


Right Triangles

Textbook: 8-1 Geometric Mean and the Pythagorean Theorem

8-2 Special Right Triangles

8-3 Ratios in Right Triangles


The Pythagorean Relationship (PDF) (Word)

Special Right Triangles and Right Triangle Trigonometry (PDF)

(Word)

Key Vocabulary

angle of depression

angle of elevation

area of a triangle

cosine

hypotenuse

Pythagorean Theorem

right triangle

sine

tangent

trigonometry

45°-45°-90° triangle

30°-60°-90° triangle

Essential Questions

How can one determine a missing measurement of a right triangle?

How can one verify that a triangle is a right triangle?

What is a trigonometric ratio?

What is the relationship between sine and cosine in terms of

complementary angles?


Right triangles (any triangle with one 90° angle) are triangles with



The side opposite the right angle in a right triangle is the hypotenuse.

It is always the longest side of a right triangle.

Special right triangles are the 30° - 60° - 90° and the 45° - 45° - 90°.

- In a 45° - 45° - 90° triangle, the hypotenuse is 2 times as long

as one of the legs.

- In the 30° - 60° - 90° triangles, the hypotenuse is twice as long

as the shorter leg and the longer leg is 3 times as long as the

shorter leg.

The Pythagorean Theorem states that in a right triangle, the square of

the measure of the hypotenuse equals the sum of the squares of the

measures of the legs. The converse of the Pythagorean Theorem states

that if the square of the measure of the longest side equals the sum of

the squares of the measures of the other two sides of a triangle, then

the triangle is a right triangle.

If the square of the longest side of a triangle is greater than the sum of

the squares of the other two sides, then the triangle is an obtuse

triangle.

If the square of the longest side of a triangle is less than the sum of the

squares of the other two sides, then the triangle is an acute triangle.

Pythagorean Triples are three positive integers that satisfy the

Pythagorean theorem.

In a right triangle with the altitude drawn to the hypotenuse, the

geometric mean can be used to find missing measures of that triangle.

(continued)





http://www.doe.virginia.gov/testing/solsearch/sol/math/G/m_ess_g-8_2.pdf

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Right Triangles (continued) Teacher Notes and Elaborations (continued)

If r, s, and t are positive numbers with r s

s t , then s is the geometric

mean between r and t. Similar right triangles have the same shape but

not necessarily the same size. They can be used to find missing

triangle segments.

Trigonometry is a branch of mathematics that combines arithmetic,

algebra, and geometry. The right triangle is the basis of trigonometry.

In any right triangle, the ratio (quotient) of the lengths of two sides is

called a trigonometric ratio. Sine is the ratio of the side opposite an

acute angle to the hypotenuse. Cosine is the ratio of the side adjacent

an acute angle to the hypotenuse. Tangent is the ratio of the side

opposite an acute angle to the adjacent side. Sine and cosine relate an

angle measure to the ratio of the measures of a triangle’s leg to its

hypotenuse. The sine of one acute angle in a right triangle and cosine

of its complement is the same.

Example:

60º 13

sin 3018

13

cos6018

13 18 sin30 = cos60

30º

The angle of elevation is the angle formed by a horizontal line and the

line of sight to an object above that horizontal line. The angle of

depression is the angle formed by a horizontal line and the line of

sight to an object below that horizontal line. The angle of elevation

and the angle of depression in the same diagram are always congruent. Return to Course Outline


Use the Law of Sines and the Law of Cosines to find missing

measures in triangles.

Find the geometric mean in right triangles.

Extension for PreAP Geometry The Law of Sines states that for any triangle with angles of measures

A, B, and C, and sides of lengths a, b, and c (a opposite A ,

opposite b B , and opposite c C ). This law is often used if two angles

and a side are known (AAS or ASA).

sin sin sinA B C

a b c

The Law of Cosines states that for any triangle with sides of lengths a,

b, and c then 2 2 2 2 cosc a b ab C . This law is often used when at least

two sides are known (SAS or SSS).

The measures of the altitude drawn from the vertex of the right angle

of a right triangle to its hypotenuse, is the geometric mean between

the measures of the two segments of the hypotenuse.

h

x h

h y

x y

If the altitude is drawn to the hypotenuse of a right triangle, then the

measure of a leg of the triangle is the geometric mean between the

measures of the hypotenuse and the segment of the hypotenuse

adjacent to that leg.

(continued)



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Right Triangles (continued) Extension for PreAP Geometry (continued)

a h b x a

a c and

y b

b c

x

c

Sample Instructional Strategies and Activities Use pieces of yarn, straws, or sticks to see which combinations of

lengths can be used to make acute, obtuse, and right triangles.

Have students make a hypsometer, then go outside and measure the

heights of buildings, trees, poles, etc., with the hypsometer.

The teacher prepares a set of clue cards containing trigonometry

word problems. Students work in groups of 4 or 5 draw a diagram

of the problem, set up a trig equation, then solve the problem.


(continued)



Polygons

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Polygons

Strand: Triangles; Polygons and

Circles

SOL G.10 The student will solve real-world

problems involving angles of

polygons.






Solve real-world problems involving the

measures of interior and exterior angles of

polygons.

Identify tessellations in art, construction, and

nature.

Find the sum of the measures of the interior

and exterior angles of a convex polygon.

Find the measure of each interior and exterior

angle of a regular polygon.

Find the number of sides of a regular polygon,

given the measures of interior or exterior

angles of the polygon.


A regular polygon will tessellate the plane if

the measure of an interior angle is a factor of

360.

Both regular and nonregular polygons can

tessellate the plane.

Two intersecting lines form angles with


An exterior angle is formed by extending a

side of a polygon.

The exterior angle and the corresponding

interior angle form a linear pair.

The sum of the measures of the interior angles

of a convex polygon may be found by

dividing the interior of the polygon into

nonoverlapping triangles.

(continued)



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Polygons (continued) Resources


Tessellations

Textbook: 4-1 Classifying Triangles

4-2 Measuring Angles in Triangles

4-6 Analyzing Isosceles Triangles

5-4 Inequalities for Sides and Angles of a Triangle

10-1 Polygons

10-2 Tessellations


How Many Triangles? (PDF) (Word)

Angles in Polygons (PDF) (Word)

Key Vocabulary

altitude

apothem

concave

convex

decagon

diagonal

dodecagon

exterior angle

heptagon

hexagon

interior angle

isosceles

linear pair

median

n-gon

nonagon

octagon

opposite

pentagon

polygon

quadrilateral

regular/irregular polygon

scalene

tessellation

tiling

triangle

Triangle Inequality Theorem


Essential Questions

What are the distinguishing characteristics of a polygon?

How do we verify that polygons can tile a plane?

What are the relationships between the sides of a polygon and the

angles of a polygon?


A polygon is a plane figure formed by coplanar segments (sides) such

that (1) each segment intersects exactly two other segments, one at

each endpoint; and (2) no two points with a common endpoint are

collinear.

Polygons are named by their number of sides and classified as convex

(a line containing a side of a polygon contains no interior points of

that polygon) or concave (a line containing a side of a polygon also

contains interior points of the polygon).

Common polygons:

3 sides: triangle 7 sides: heptagon 10 sides: decagon

4 sides: quadrilateral 8 sides: octagon 12 sides: dodecagon

5 sides: pentagon 9 sides: nonagon n sides: n-gon

6 sides: hexagon

A segment joining two nonconsecutive vertices is a diagonal of the

polygon.

Two angles that are adjacent (share a leg) and supplementary (add up

to 180°) form a linear pair.

(continued)








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Polygons (continued) Teacher Notes and Elaborations (continued)

Polygons have interior angles (angles formed by the sides of the

polygon and enclosed by the polygon) and exterior angles (angles

formed by extending an existing side). The exterior angle and the

corresponding interior angle form a linear pair. The sum of the

measures of the interior angles of a polygon is found by multiplying

two less than the number of sides by 180°, [ ( 2)180n ]. The sum of

the measures of the exterior angles, one at each vertex, is 360°.

A regular polygon is a convex polygon with all sides congruent and

all angles congruent. The center of a regular polygon is the center of

the circumscribed circle. Given the measure of an exterior angle of a

regular polygon, the number of sides can be determined by dividing

360° by the measure of that angle. The central angle of a regular

polygon is an angle formed by two radii drawn to consecutive

vertices. Its measure can be determined by dividing 360° by the

number of sides.

A polygon will tessellate the plane if the interior angles at a vertex add

to 360°. Tessellations are repeated copies of a figure that completely

fill a plane without overlapping. The hexagon pattern in a honeycomb

is a tessellation of regular hexagons. Both regular and non-regular

polygons can tessellate the plane.


When a tessellation uses only one shape it is called a pure tessellation.

The three regular polygons that create pure tessellations are triangle,

square, and hexagon.

Regular polygon tessellation Non-regular polygon tessellation


Investigate and identify the regular polygons that tessellate.

Distinguish between pure and semi-pure tessellations.


Tessellations that involve more than one type of shape are called

semi-pure tessellations. For example, in an octagon – square

tessellation, two regular octagons, and a square meet at each vertex

point.

Students, using materials of their choice, will make mobiles with

different polygons.

Students bring in photographs of regular polygons in art, nature, or

architecture.

Find tessellations in real world situations such as in art and

architecture.

Pattern blocks may be used to create tessellations.

Students can design a book cover using tessellations

(continued)



Quadrilaterals

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Parallelograms

Rectangles

Rhombi and Squares

Trapezoids and Kites

Strand: Polygons and Circles

SOL G.9 The student will verify characteristics

of quadrilaterals and use properties of

quadrilaterals to solve real-world

problems.






Solve problems, including real-world

problems, using the properties specific to

parallelograms, rectangles, rhombi, squares,

isosceles trapezoids, and trapezoids.

Prove that quadrilaterals have specific

properties, using coordinate and algebraic

methods, such as the distance formula, slope,

and midpoint formula.

Prove the characteristics of quadrilaterals,

using deductive reasoning, algebraic, and

coordinate methods.

Prove properties of angles for a quadrilateral

inscribed in a circle.


The terms characteristics and properties can

be used interchangeably to describe

quadrilaterals. The term characteristics is

used in elementary and middle school

mathematics.

Quadrilaterals have a hierarchical nature

based on the relationships between their sides,

angles, and diagonals.

Characteristics of quadrilaterals can be used to

identify the quadrilateral and to find the

measures of sides and angles.

(continued)



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Quadrilaterals (continued) Resources


Quadrilaterals

Textbook: 6-1 Parallelograms

6-2 Tests for Parallelograms

6-3 Rectangles

6-4 Squares and Rhombi

6-4B Extension - Kites

6-5 Trapezoids


Properties of Quadrilaterals (PDF) (Word)

Key Vocabulary

base

base angles

characteristics

diagonal

isosceles trapezoid

kite

legs

median of a trapezoid

parallelogram

quadrilateral

rectangle

rhombus

square

trapezoid

Essential Questions

What are the distinguishing features of the different types of

quadrilaterals?

How are the properties of quadrilaterals used to solve real-life

problems?

What is the hierarchical nature among quadrilaterals?



Algebraic methods and coordinate methods such as distance formula,

midpoint formula, and the slope formula can be used to prove

quadrilateral properties.

A quadrilateral is a polygon with four sides. Quadrilaterals have a

hierarchical nature based on relationships among their sides, their

angles, and their diagonals. The diagonal of a polygon is a segment

joining two nonconsecutive vertices of the polygon.

A parallelogram is a quadrilateral with opposite sides parallel and

congruent. Consecutive angles of a parallelogram are supplementary;

opposite angles are congruent; and the diagonals of a parallelogram

bisect each other.

A rectangle is a parallelogram with four right angles. The diagonals of

a rectangle are congruent.

A rhombus is a parallelogram with congruent sides. The diagonals of a

rhombus are perpendicular and bisect each other and the opposite

angles.

A square is a parallelogram, a rectangle, and a rhombus.

A trapezoid is a quadrilateral with exactly one pair of opposite sides

parallel. An isosceles trapezoid has congruent legs (the non-parallel

sides). Both pairs of base angles in an isosceles trapezoid are

congruent and diagonals are congruent. The median of a trapezoid is

the segment that joins the midpoints of the legs. It is parallel to the

bases and has a length equal to half the sum of the lengths of the bases.

A kite is a quadrilateral with two pairs of congruent adjacent sides. (continued)






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Quadrilaterals (continued) Teacher Notes and Elaborations (continued) Characteristics of quadrilaterals are used to identify figures, and to find

values for missing parts and areas.

Areas of work that use quadrilaterals include art, construction, fabric

design, and architecture.

The hierarchical nature of quadrilaterals can be described as ranking

based on characteristics.

If a quadrilateral is inscribed in a circle, its opposite angles are

supplementary. This can be verified by considering that the arcs

intercepted by opposite angles of an inscribed quadrilateral form a

circle.


Example:

Quadrilateral ABCD is inscribed in a circle.

AB BC CD DA 360m m m m

The measure of 1

DAB = BCD2

m m and the measure of

1

BCD = DAB2

m m . A B

BCD = 2 A and DAB = 2 Cm m m m

BCD DAB 360m m

2 A+2 C = 360m m D C

A C 180m m

Sample Instructional Strategies and Activities Use flowcharts or Venn diagrams to show relationships and

properties of quadrilaterals.

Use patty paper to show properties of the different quadrilaterals.

Use notecards to create models of different quadrilaterals. Discuss

the characteristics and have students record their findings on the

back of the models.



Circles

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Terminology

Area and Circumference

Tangents

Arcs and Circles

Angle Formula

Segment Formula

Strand: Polygons and Circles

SOL G.11 The student will use angles, arcs,

chords, tangents, and secants to

a) investigate, verify, and apply

properties of circles;

b) solve real-world problems

involving properties of circles; and

c) find arc lengths and areas of

sectors in circles.






Find lengths, angle measures, and arc

measures associated with

– two intersecting chords;

– two intersecting secants;

– an intersecting secant and tangent;

– two intersecting tangents; and

– central and inscribed angles.

Calculate the area of a sector and the

length of an arc of a circle, using

proportions.

Solve real-world problems associated with

circles, using properties of angles, lines,

and arcs.

Verify properties of circles, using

deductive reasoning, algebraic, and

coordinate methods.


Many relationships exist between and among

angles, arcs, secants, chords, and tangents of a

circle.

All circles are similar.

A chord is part of a secant.

Real-world applications may be drawn from

architecture, art, and construction.

(continued)



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Circles (continued) Resources


Circles

Textbook: 9-1 Exploring Circles

9-2 Angles and Arcs

9-3 Arcs and Chords

9-4 Inscribed Angles

9-5 Tangents

9-6 Secants, Tangents, and Angle Measures

9-7 Special Segments in a Circle


Angles, Arcs, and Segments in Circles (PDF) (Word)

Arc Length and Area of a Sector (PDF) (Word)

Key Vocabulary

arc

arc length

arc measure

central angle

chord

circle

circumscribed

circumference

common tangent

concentric circles

diameter

inscribed

intercepted arc

major arc

minor arc

point of tangency

radius

secant

sector

semicircles

tangent

Essential Questions

How might geometric objects (points, segments, lines, etc.)

interact/intersect with circles? Return to Course Outline

What is area/circumference and how is it measured? What does the

value pi represent?

How does a tangent line relate to the circle?

How are the angle formulas of circles related to similar triangles?

What are the relationships between chords and arcs?

What is the difference between arc length and arc measure?


A circle is the set of all points equidistant from a given point in a

plane. The distance from the center of the circle to a point on the circle

is the radius.

The arc measure is the degree measure of its central angle. A central

angle is an angle with its vertex at the circle’s center. A central angle

separates a circle into two arcs called a major arc (measures greater

than 180º but less than 360º), and a minor arc (measures greater than

0º but less than 180º). Semicircles are the two arcs of a circle that are

cut off by a diameter. A semicircle measures 180º. An arc is an

unbroken part of a curve of a circle. The central angle measures the

same as its intercepted arc. The intercepted arc is the part of the circle

that lies between the two lines that intersect the circle.

A chord is a segment joining two points on the circle. A diameter is a

chord that passes through the circle’s center. A secant is a line that

contains a chord. A tangent is a line that intersects a circle in only one

point. Measures of chords, secant segments, and tangent segments can

be determined.

An inscribed angle is an angle whose vertex is on the circle and

whose sides are chords of the circle. The measure of an inscribed

angle is equal to one-half the measure of its intercepted arc.

(continued)



http://www.doe.virginia.gov/testing/solsearch/sol/math/G/m_ess_g-11a.pdf

http://www.doe.virginia.gov/testing/solsearch/sol/math/G/m_ess_g-11a.doc

http://www.doe.virginia.gov/testing/solsearch/sol/math/G/m_ess_g-11bc.pdf

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Circles (continued) Teacher Notes and Elaborations (continued) The measure of an angle formed by two chords that intersect inside a

circle is equal to half the sum of the measures of the intercepted arcs.

The measure of an angle formed by a chord and a tangent is equal to

half the measure of the intercepted arc.

The measure of an angle formed by two secants, two tangents, or a

secant and a tangent drawn from a point outside a circle is equal to

half the difference of the measures of the intercepted arcs.

Use the properties of chords, secants, and tangents to determine

missing lengths.

When two chords intersect inside a circle, the product of the lengths of

the segments of one chord equals the product of the lengths of the

segments of the other chord.

When two secant segments are drawn to a circle from an exterior

point, the product of the lengths of one secant segment and its exterior

segment is equal to the product of the lengths of the other secant

segment and its exterior segment.

When a tangent segment and a secant segment are drawn to a circle

from an exterior point, the square of the length of the tangent segment

is equal to the product of the lengths of the secant segment and its

exterior segment.

The length of an arc (arc length) is a linear measure and is part of the

circumference (perimeter of a circle). A sector of a circle is that part

of the circle bounded by two radii and an arc. Length of an arc and

area of a sector can be calculated using the following formulas:

In circle O, the measure of AB x (This is a degree measure.) Return to Course Outline

Length of AB 2360

xr (This is a linear measure.)

Area of sector 2AOB360

xr

Verifying the properties of circles may include definitions, postulates,

theorems, algebraic methods, and coordinate methods.

Wheels and gears are two important applications of circles.

In the same circle or congruent circles:

- Congruent chords have congruent arcs and vice versa.

- Congruent chords are equidistant from the center and vice versa.

- A diameter that is perpendicular to a chord bisects the chord and

its arc.

An angle inscribed in a semi-circle is a right angle. Opposite angles of

an inscribed quadrilateral are supplementary.


Find the area of a segment of a circle.

Find the area of an annulus.


A segment of a circle is the region between an arc and a chord of a

circle.

To find the area of a segment, find the area of

the sector and subtract the area of the triangle.

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Circles (continued) Extension for PreAP Geometry (continued)

An annulus is the region between two concentric circles.

To find the area of an annulus, find the area of the larger circle and

subtract the area of the smaller circle.

Sample Instructional Strategies and Activities Use the graphing calculator to show that a triangle inscribed in a

semicircle is a right triangle; to show that the product of the parts

of one chord equal the product of the parts of the other chord; to

graph and identify circles as tangent, intersecting, or concentric;

and to graph and recognize tangents as internal or external.

Use patty paper to demonstrate the properties of circles.

Students use post-it notes to identify intercepted arcs.

Students use post-it notes to find multiple angles and arc measures

in circle drawings.




Surface Area and Volume

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Lessons

Area of 2-D Shapes (optional)

Prisms and Pyramids

Cylinders and Cones

Spheres

Similar Objects

Strand: Three-Dimensional Figures

SOL G.13

The student will use formulas for

surface area and volume of three-

dimensional objects to solve real-

world problems.

SOL G.14

The student will use similar geometric

objects in two- or three-dimensions to

a) compare ratios between side

lengths, perimeters, areas, and

volumes;

b) determine how changes in one or

more dimensions of an object affect

area and/or volume of the object;

c) determine how changes in area

and/or volume of an object affect

one or more dimensions of the

object; and

d) solve real-world problems about

similar geometric objects.






Find the total surface area of cylinders, prisms,

pyramids, cones, and spheres, using the

appropriate formulas.

Calculate the volume of cylinders, prisms,

pyramids, cones, and spheres, using the

appropriate formulas.

Solve problems, including real-world

problems, involving total surface area and

volume of cylinders, prisms, pyramids, cones,

and spheres as well as combinations of three-

dimensional figures.

Calculators may be used to find decimal

approximations for results.

Compare ratios between side lengths,

perimeters, areas, and volumes, given two

similar figures.

Describe how changes in one or more

dimensions affect other derived measures

(perimeter, area, total surface area, and

volume) of an object.

Describe how changes in one or more

measures (perimeter, area, total surface area,

and volume) affect other measures of an

object.

Solve real-world problems involving measured

attributes of similar objects.


The surface area of a three-dimensional object

is the sum of the areas of all its faces.

The volume of a three-dimensional object is

the number of unit cubes that would fill the

object.

A change in one dimension of an object

results in predictable changes in area and/or

volume.

A constant ratio exists between corresponding

lengths of sides of similar figures.

Proportional reasoning is integral to

comparing attribute measures in similar

objects.

(continued)



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Surface Area and Volume (continued) Resources


Surface Area and Volume

Similar Geometric Objects

Textbook: 11-1 Exploring Three-Dimensional Figures

11-2 Nets and Surface Area

11-3 Surface Area of Prisms and Cylinders

11-4 Surface Area of Pyramids and Cones

11-5 Volume of Prisms and Cylinders

11-6 Volume of Pyramids and Cones

11-7 Surface Area and Volume of Spheres

11-8 Congruent and Similar Solids


Surface Area and Volume (PDF) (Word)

Similar Solids and Proportional Reasoning (PDF) (Word)

Key Vocabulary

altitude

area

base

base area (B)

base edge

cone

cube

face

height

lateral edge

lateral area

prism

polygon

pyramid

similar figures

slant height

sphere

surface area

three-dimensional

two dimensional

vertex

volume

Essential Questions

What is area?

What is volume?

How are the lateral area, surface area, and volume of the following

figures determined: prisms, cylinders, pyramids, cones, and

spheres? Return to Course Outline

How does a change in dimensions affect the area and/or volume of

the object?

In similar figures, how does a change of one measurement affect

perimeter, area, or volume?


A dimension is the number of coordinates required to locate a point in

a space. A flat surface is two-dimensional because two coordinates are

needed to specify a point on it. Three-dimensional space is a

geometric model of the physical universe in which we live. The three

dimensions are commonly called length, width, and depth (or height),

although any three directions can be chosen, provided that they do not

lie in the same plane.

A polygon is a geometric figure formed by three or more coplanar

segments called sides. Each side intersects exactly two other sides, but

only at their endpoints, and the intersecting sides must be

noncollinear.

A vertex of an angle is a point common to the two sides of the angle.

In a polygon, a vertex is a point common to two sides of the polygon.

The vertex of a polyhedron is a point common to the edges of a

polyhedron. In a polyhedron the flat surfaces formed by the polygons

and their interiors are called faces.

Area is the number of square units in a region. Surface area is a

measurement of coverage such as wallpaper.

Lateral area is the area of the exterior surface (lateral surface) of a

three-dimensional figure not including the area of the base(s).

A prism is a three-dimensional figure whose lateral faces are (continued)









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Surface Area and Volume (continued) Teacher Notes and Elaborations (continued) parallelograms. If the faces are rectangles, the prism is a right prism.

A prism is classified by the shape of its base.

A pyramid is a three-dimensional figure whose lateral faces are

triangles. In regular pyramids, the base is a regular polygon, lateral

edges are congruent, and all lateral faces are congruent isosceles

triangles. Slant height in a pyramid is the distance from the vertex

perpendicular to the base on a lateral face of the pyramid. Slant height

on a cone is the distance from the vertex to the circle. Height is the

perpendicular distance between bases or between a vertex and a base.

A cone is a three-dimensional figure that has a circular base, a vertex

not in the plane of the circle, and a curved lateral surface. In a right

cone, the altitude is a perpendicular segment from the vertex to the

center of the base. The height (h) is the length of the altitude. The

slant height ( ) is the distance from the vertex to a point on the edge

of the base.

Surface area is the lateral area plus the area of the base(s). Bases of

prisms are congruent polygons lying in parallel planes. An altitude

(height) of a prism is a segment joining the two base planes and

perpendicular to both. The faces of a prism that are not its bases are

called lateral faces. Adjacent lateral faces intersect in parallel

segments called lateral edges. In right prisms the lateral edges are also

altitudes.

Volume is the capacity of a three-dimensional figure such as the

amount of water in an aquarium.

The volume of an irregularly shaped object can be found by

measuring its displacement. When an object is placed in a liquid, it

causes the liquid to rise. This volume is called the objects’

displacement. Return to Course Outline

The base of a three-dimensional figure could be a circle, a triangle, a

square, a rectangle, a regular hexagon or another type of polygon.

Many formulas use B to represent the area of the base of the solid

figure. To find the area of a base (B) in three dimensional figures, use

the area formula that applies. Formulas for those figures may need to

be reviewed.

A sphere is the set of all points in space equidistant from a given

point. The center is the given point and the radius is the given

distance. Surface area and volume of spheres will also be found.

When determining surface area of combinations of solids, attention

needs to be given to the possibility of shared faces.

Similar figures are figures that have the same shape but not

necessarily the same size.

Scale factors (proportional reasoning) are used to compare perimeters,

areas, and volumes of similar two-dimensional and three-dimensional

geometric figures. A change in one dimension of an object results in

changes in area and volume in specific patterns.

Volumes, areas, and perimeters of similar polygons are examined to

draw conclusions about how changes in one dimension affect both

area and volume.

If the given perimeter of a polygon is increased or decreased, the area

will increase or decrease by the square of the change and the volume

increases or decreases by the cube of the change.

Similar solids are solids that have the same shape but not necessarily

the same size. All spheres are similar.

(continued)



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Surface Area and Volume (continued) Teacher Notes and Elaborations (continued) If the scale factor of two similar solids is a:b, then:

– The ratio of corresponding perimeters is a:b.

– The ratios of the base areas, of the lateral areas, and of the total

areas are a2:b2.

– The ratio of the volumes is a3:b3.


Use strings, straws, toothpicks, etc. to make three-dimensional

objects.

Students make a three-dimensional object from any material they

choose. They calculate lateral area, total area, and volume and

incorporate this into a written report, which includes their

calculations, a sketch of their model, and a description of their

procedure. Students give a brief oral report of their project.

Using a geometric model kit, students will investigate relationships

among volume formulas.

Demonstrate a way that the formula for the surface area of a sphere

might have been evolved.

To demonstrate the formula for surface area of a sphere, cut an

orange in half and trace the circumference of the orange on paper

several times. Peel the orange and completely fill as many circles

as possible. The result should be four filled circles, thus four times

the area of the circle.

Using items from a pantry have students measure and compute

surface area and volume.

When an object is placed in a liquid, it causes the liquid to rise.

This volume is called the objects’ displacement. The volume of an

irregularly shaped object can be found by measuring its

displacement.


Example: A rock is placed into a rectangular prism containing water.

The base of the container is 10 centimeters by 15 centimeters and

when the rock is put in the prism, the water level rises 2 centimeters

due to the displacement. This new “slice” of water has a volume of

300 cubic centimeters (10 15 2 ). Therefore, the volume of the rock is

300 cubic centimeters.

Each student is given a sheet of construction paper. Next, they are

instructed to cut a square from each corner and form an open top

box with the maximum volume.

Have students use string and a ruler to determine whether two

solids are similar. If the figures are similar then use the

measurements to compare areas and volumes.



Coordinate Geometry

Fou

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g P

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d

Lessons

Formulas

Quadrilaterals

Transformations

o Including dilations

Writing the Equation of a Circle

Strand: Reasoning, Lines, and

Transformations; Polygons and Circles

SOL G.3 The student will use pictorial representations,

including computer software, constructions,

and coordinate methods, to solve problems

involving symmetry and transformation. This

will include

a) investigating and using formulas for

finding distance, midpoint, and slope;

b) applying slope to verify and determine

whether lines are parallel or

perpendicular;

c) investigating symmetry and determining

whether a figure is symmetric with

respect to a line or a point; and

d) determining whether a figure has been

translated, reflected, rotated, or dilated,

using coordinate methods.

SOL G.9 The student will verify characteristics of

quadrilaterals and use properties of

quadrilaterals to solve real-world problems.

SOL G.12 The student, given the coordinates of the

center of a circle and a point on the circle, will

write the equation of the circle.



The student will use problem solving, mathematical

communication, mathematical reasoning, connections, and

representations to

Find the coordinates of the midpoint of a segment, using

the midpoint formula.

Use a formula to find the slope of a line.

Compare the slopes to determine whether two lines are

parallel, perpendicular, or neither.

Determine whether a figure has point symmetry, line

symmetry, both, or neither.

Given an image and pre-image, identify the

transformation that has taken place as a reflection,

rotation, dilation, or translation.

Apply the distance formula to find the length of a line

segment when given the coordinates of the endpoints.

Solve problems, including real-world problems, using

the properties specific to parallelograms, rectangles,

rhombi, squares, isosceles trapezoids, and trapezoids.

Prove that quadrilaterals have specific properties, using

coordinate and algebraic methods, such as the distance

formula, slope, and midpoint formula.

Prove the characteristics of quadrilaterals, using

deductive reasoning, algebraic, and coordinate methods.

Prove properties of angles for a quadrilateral inscribed

in a circle.

Identify the center, radius, and diameter of a circle from

a given standard equation.

Use the distance formula to find the radius of a circle.

Given the coordinates of the center and radius of the

circle, identify a point on the circle.

Given the equation of a circle in standard form, identify

the coordinates of the center and find the radius of the

circle.

Given the coordinates of the endpoints of a diameter,

find the equation of the circle.

Given the coordinates of the center and a point on the

circle, find the equation of the circle.

Recognize that the equation of a circle of given center

and radius is derived using the Pythagorean Theorem.

Essential Understanding Transformations and combinations of transformations

can be used to describe movement of objects in a plane.

The distance formula is an application of the

Pythagorean Theorem.

Geometric figures can be represented in the coordinate

plane.

Techniques for investigating symmetry may include

paper folding, coordinate methods, and dynamic

geometry software.

Parallel lines have the same slope.

The product of the slopes of perpendicular lines is -1.

The image of an object or function graph after an

isomorphic transformation is congruent to the preimage

of the object.

The terms characteristics and properties can be used

interchangeably to describe quadrilaterals. The term

characteristics is used in elementary and middle school

mathematics.

Quadrilaterals have a hierarchical nature based on the

relationships between their sides, angles, and diagonals.

Characteristics of quadrilaterals can be used to identify

the quadrilateral and to find the measures of sides and

angles.

A circle is a locus of points equidistant from a given

point, the center.

Standard form for the equation of a circle is

2 2 2x h y k r , where the coordinates of the

center of the circle are ( , )h k and r is the length of the

radius.

The circle is a conic section.

(continued)



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Coordinate Geometry (continued) Resources


G.3 - Symmetry & Transformations

G.9 - Quadrilaterals

G.12 – Equation of a Circle

Textbook: 1-4 Measuring Segments

1-5 Midpoints and Segment Congruence

3-3 Slopes of Lines

Ch. 6 Exploring Quadrilaterals

13-4 Mappings

13-5 Reflections

13-6 Translations

13-7 Rotations

13-8 Dilations


Distance and Midpoint Formulas (PDF) (Word)

Slope (PDF) (Word)

Symmetry (PDF) (Word)

Transformations (PDF) (Word)

Circles in the Coordinate Plane (PDF) (Word)

Key Vocabulary

dilation

distance formula

image

isometry

isomorphism

line symmetry

midpoint formula

point symmetry

pre-image

reflection

rotation

slope

slope formula

standard form for the

equation of a circle

symmetry

transformation

translation

Essential Questions

What is the relationship between the distance formula, the

Pythagorean Theorem, and the equation of a circle? Return to Course Outline

How does the concept of midpoint and slope relate to symmetry

and transformation?

What is line symmetry?

What is point symmetry?

How can symmetry be used to describe naturally occurring

phenomena?

How is a figure translated, reflected, rotated, or dilated?

What is the relationship between the center, the radius, and the

standard equation of a circle?


Like finding distance, two situations must be considered to find the

midpoint of the line and the congruence of the two line segments. The

two situations that must be considered are the midpoint on a number

line and midpoint in the coordinate plane. The midpoint of a segment

is the point that divides the segment into two congruent segments. The

midpoint of AB is the average of the coordinates of A and B.

A M B

3 2 1 0 1 2 3 4 5 6 7

( 1) 5

22

The Midpoint Formula uses the idea that the midpoint of a horizontal

or vertical line is the average of the coordinates of the endpoints. To

find the midpoint of a horizontal line segment, find the average of the

x endpoint coordinates; the y coordinate will be the same for all the

points. To find the midpoint of a vertical line segment the x

coordinate; will be the same for all points; the y coordinate will be the

average of the y endpoint coordinates.

(continued)




http://www.doe.virginia.gov/testing/solsearch/sol/math/G/m_ess_g-3a.pdf

http://www.doe.virginia.gov/testing/solsearch/sol/math/G/m_ess_g-3a.doc

http://www.doe.virginia.gov/testing/solsearch/sol/math/G/m_ess_g-3ab.pdf

http://www.doe.virginia.gov/testing/solsearch/sol/math/G/m_ess_g-3ab.doc

http://www.doe.virginia.gov/testing/solsearch/sol/math/G/m_ess_g-3c.pdf

http://www.doe.virginia.gov/testing/solsearch/sol/math/G/m_ess_g-3c.doc

http://www.doe.virginia.gov/testing/solsearch/sol/math/G/m_ess_g-3d.pdf

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Coordinate Geometry (continued) Teacher Notes and Elaborations (continued)

-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

-9

-8

-7

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

8

9

x

y

C D E

-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

-9

-8

-7

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

8

9

x

y

F

G

H

The midpoint of CE is D (2,2) . The midpoint of FH is G ( 3, 2)

.

This idea is used twice to find the coordinates of the midpoint of a

slanting segment with endpoints 1 1 1P ( , )x y and

2 2 2P ( , )x y .

-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

-9

-8

-7

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

8

9

x

y

P1(x1, y1)

P2(x2, y2)

M S

R T

The midpoint of 1 2P P is M 1 2 1 2,

2 2

x x y y

.

Some students may have difficulty in extending the concept of finding

the midpoint of a line segment on one number line to a line segment in

the coordinate plane. Using models such as the one above will aid in

developing this concept. Return to Course Outline

The slope (effect of steepness) of a line containing two points in the

coordinate plane can be found using the slope formula. The slope of a

vertical line is undefined since x1 = x2. Parallel lines are lines that do

not intersect and are coplanar. Parallel planes are planes that do not

intersect. Nonvertical lines are parallel if they have the same slope and

different y-intercepts. Any two vertical lines are parallel.

Perpendicular lines are lines that intersect at right angles. Two non-

vertical lines are perpendicular if and only if the product of their

slopes is 1 .

Students should have multiple experiences applying the following

formulas.

Given two points (x1, y1) and (x2, y2):

- the midpoint formula is 1 2 1 2,2 2

x x y y

;

- the distance formula is 2 2

2 1 2 1x x y y ; and

- the slope formula is

2 1

2 1

y y

x x

.

Regular polygons are frequently used to introduce the concepts of

symmetry, transformations, and tessellation. A geometric

configuration (curve, surface, etc.) is said to be symmetric (have

symmetry) with respect to a point, a line, or a plane, when for every

point on the configuration there is another point of the configuration

such that the pair is symmetric with respect to the point, line, or plane.

The point is the center of symmetry; the line is the axis of symmetry,

and the plane is the plane of symmetry. A line of symmetry is a line

(continued)



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Coordinate Geometry (continued) Teacher Notes and Elaborations (continued) that can be drawn so that the figure on one side is the reflection image

of the figure on the opposite side.

A figure has point symmetry if there is a symmetry point O such that

the half-turn HO maps the figure onto itself. A figure has line

symmetry if there is a symmetry line k such that the reflection Rk maps

the figure onto itself.

An isomorphism is a one-to-one mapping that preserves the

relationship between two sets. The original figure is the preimage. The

resulting figure is an image. An isometry is a transformation in which

the preimage and image are congruent. Reflections, rotations, and

translations are isometries. Dilations are not isometry.

Reflection is a transformation in which a line acts like a mirror,

reflecting points to their images. For many figures, a point can be

found that is a point of reflection for all points on the figure. This

point of reflection is called a point of symmetry.

A rotation is a transformation suggested by a rotating paddle wheel.

When the wheel moves, each paddle rotates to a new position. When

the wheel stops, the position of a paddle ( P ) can be referred to

mathematically as the image of the initial position of the paddle (P). A

figure with rotational symmetry of 180° has point symmetry.

A geometric transformation in a plane is a one-to-one correspondence

between two sets of points. It is a change in its position, shape, or size.

It maps a figure onto its image and may be described with arrow (→)

notation. A reflection is a type of transformation that can be described

by folding over a line of reflection or line of symmetry. For some

figures, a point can be found that is a point of reflection for all points

on the figure.


A dilation is a transformation that may change the size of a figure. It

requires a center point and a scale factor. The scale factor is defined as

the image to pre-image. For example: 4 to 3 or 4

3 represents an

enlargement.

A composite of reflections is the transformation that results from

performing one reflection after another. A translation (slide) is the

composite of two reflections over parallel lines. The Pythagorean Theorem (distance formula) can be used to develop

an equation of a circle.

y

Let P(x, y) represent any point on the circle.

The distance between C(h, k) and P(x, y) is r.

2 2( ) ( )x h y k r P(x, y)

2 2 2( ) ( )x h y k r C(h, k)

x

Given the coordinates of the center of the circle (h, k) and a radius r,

four easily identified points on the circle are:

( , )h r k , ( , )h r k , ( , )h k r , ( , )h k r

Given the coordinates of the endpoints of a diameter, midpoint

formula can be used to find the center of the circle and distance

formula can be used to find the radius

(continued)

r



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Coordinate Geometry (continued) Extension for PreAP Geometry

Reflect triangles over horizontal and vertical lines in the coordinate

plane and the line y = x.

Draw on a coordinate plane the image that results from a geometric

figure that has been reflected, rotated, or dilated.

Investigate the relationship between a rotation and the composition

of reflections.

Investigate point-slope form as it relates to the equation of a line

(slope-intercept form) and the formula for slope.

Use slopes of parallel and perpendicular lines to write equations in

standard, point-slope, and slope-intercept forms.

Find the coordinates of an endpoint of a segment given the

coordinates of the midpoint and one endpoint.


Point-slope form is an equation of the form 1 1( )y y m x x for the

line passing through a point whose coordinates are 1 1( , )x y and having

slope m.

The composite of reflections with respect to two intersecting lines is a

transformation called a rotation. The point of intersection, point P, is

the center of rotation. The figure rotates or turns around the point P.

Point symmetry is a rotational symmetry of 180°.

A dilation is a similarity transformation that alters the size of a

geometric figure, but does not change the shape. For each dilation, a

scale factor enlarges the dilation image, reduces the dilation image, or

maintains a congruence transformation.


Investigate and identify points that lie inside, on, or outside a

circle.

Write inequality statements for regions either inside or outside a Return to Course Outline

circle and sketch these graphs.


An example of an inequality that describes the points (x, y) outside the

circle that are more than three units from center (4, 2 ) is 2 2( 4) ( 2) 9x y . The graph would be a broken circle and shaded

outside the circle.

An example of an inequality that describes the points (x, y) inside the

circle that are less than or equal to four units from center ( 3, 5 ) is 2 2( 3) ( 5) 16x y . The graph would be a circle and shaded inside the

circle.


Do activities from the Geometer’s Sketchpad by Key Curriculum

Press.

Use coordinate geometry as a tool for making conjectures about

midpoints, slopes, and distance.

Each student is given a sheet of construction paper. Next, the

teacher puts a few drops of finger paint, etc. on each paper. Each

student folds his/her papers to illustrate symmetry with respect to a

line.

Demonstrate symmetry by using patty paper.

Cut out a triangle. Place a different color dot in each angle. Place

the triangle on the paper and trace around it in pencil. Slide

triangle over and mark the color in each angle so that the colors

correspond with the cardboard triangle. Place triangle back on top

and rotate it so that it no longer overlaps. Repeat until the plane is

filled. Have students identify parallel lines, vertical angles, etc.

Students make conjectures about lines and angles in the

tessellation. Students are given various polygons and asked if they

tessellate a plane. Explain why or why not. (continued)



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Coordinate Geometry (continued) Sample Instructional Strategies and Activities (continued)

Place a shape on the overhead projector. Have a student trace the

image on the blackboard. Move the projector away from the board

and trace the new image. Take the original shape and compare the

angles of the original with the angles of the images. Students can

measure the lengths of the sides and compare ratios.

Use patty paper to demonstrate reflections, rotations, dilations, or

translations.

Use examples of advertisements to identify examples of

transformations.

Give students coordinates of the vertices of a rectangle. Have

students find the lengths of the diagonals, the midpoints of the

diagonals, and the slopes of the diagonals. Have students make

conjectures about the diagonals of the rectangle. Repeat with

square, rhombus, parallelogram, isosceles trapezoid, trapezoid,

and quadrilateral. Have students make conjectures about the

diagonals of each.


http://www.doe.virginia.gov/testing/test_administration/ancilliary_materials/mathematics/2009/2009_sol_formula_sheet_geometry.pdf

http://www.doe.virginia.gov/testing/test_administration/ancilliary_materials/mathematics/2009/2009_sol_formula_sheet_geometry.pdf


Geometry

Copyright © 2009

by the

Virginia Department of Education

P.O. Box 2120

Richmond, Virginia 23218-2120

http://www.doe.virginia.gov

All rights reserved. Reproduction of these materials for instructional purposes in public school classrooms in Virginia is permitted.

Superintendent of Public Instruction

Patricia I. Wright, Ed.D.

Assistant Superintendent for Instruction

Linda M. Wallinger, Ph.D.

Office of Elementary Instruction

Mark R. Allan, Ph.D., Director

Deborah P. Wickham, Ph.D., Mathematics Specialist

Office of Middle and High School Instruction

Michael F. Bolling, Mathematics Coordinator

Acknowledgements

The Virginia Department of Education wishes to express sincere thanks to Deborah Kiger Bliss, Lois A. Williams, Ed.D., and Felicia Dyke, Ph.D.

who assisted in the development of the 2009 Mathematics Standards of Learning Curriculum Framework.

NOTICE

The Virginia Department of Education does not unlawfully discriminate on the basis of race, color, sex, national origin, age, or disability in

employment or in its educational programs or services.

The 2009 Mathematics Curriculum Framework can be found in PDF and Microsoft Word file formats on the Virginia Department of Education’s

Web site at http://www.doe.virginia.gov.

http://doe.virginia.gov/

http://www.doe.virginia.gov/

Virginia Mathematics Standards of Learning Curriculum Framework 2009

Introduction

The 2009 Mathematics Standards of Learning Curriculum Framework is a companion document to the 2009 Mathematics Standards of Learning and

amplifies the Mathematics Standards of Learning by defining the content knowledge, skills, and understandings that are measured by the Standards

of Learning assessments. The Curriculum Framework provides additional guidance to school divisions and their teachers as they develop an

instructional program appropriate for their students. It assists teachers in their lesson planning by identifying essential understandings, defining

essential content knowledge, and describing the intellectual skills students need to use. This supplemental framework delineates in greater specificity

the content that all teachers should teach and all students should learn.

Each topic in the Mathematics Standards of Learning Curriculum Framework is developed around the Standards of Learning. The format of the

Curriculum Framework facilitates teacher planning by identifying the key concepts, knowledge and skills that should be the focus of instruction for

each standard. The Curriculum Framework is divided into two columns: Essential Understandings and Essential Knowledge and Skills. The purpose

of each column is explained below.

Essential Understandings

This section delineates the key concepts, ideas and mathematical relationships that all students should grasp to demonstrate an understanding of the

Standards of Learning.


Each standard is expanded in the Essential Knowledge and Skills column. What each student should know and be able to do in each standard is

outlined. This is not meant to be an exhaustive list nor a list that limits what is taught in the classroom. It is meant to be the key knowledge and skills

that define the standard.

The Curriculum Framework serves as a guide for Standards of Learning assessment development. Assessment items may not and should not be a

verbatim reflection of the information presented in the Curriculum Framework. Students are expected to continue to apply knowledge and skills

from Standards of Learning presented in previous grades as they build mathematical expertise.

Mathematics Standards of Learning Curriculum Framework 2009: Geometry 1

TOPIC: REASONING, LINES, AND TRANSFORMATIONS

GEOMETRY

STANDARD G.1

The student will construct and judge the validity of a logical argument consisting of a set of premises and a conclusion. This will include

a) identifying the converse, inverse, and contrapositive of a conditional statement;

b) translating a short verbal argument into symbolic form;

c) using Venn diagrams to represent set relationships; and


ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS

Inductive reasoning, deductive reasoning, and proof are critical in

establishing general claims.

Deductive reasoning is the method that uses logic to draw

conclusions based on definitions, postulates, and theorems.

Inductive reasoning is the method of drawing conclusions from a

limited set of observations.

Proof is a justification that is logically valid and based on initial

assumptions, definitions, postulates, and theorems.

Logical arguments consist of a set of premises or hypotheses and

a conclusion.

Euclidean geometry is an axiomatic system based on undefined

terms (point, line and plane), postulates, and theorems.

When a conditional and its converse are true, the statements can

be written as a biconditional, i.e., iff or if and only if.

Logical arguments that are valid may not be true. Truth and

validity are not synonymous.

The student will use problem solving, mathematical communication,

mathematical reasoning, connections, and representations to

Identify the converse, inverse, and contrapositive of a conditional

statement.

Translate verbal arguments into symbolic form, such as

(p q) and (~p ~q).

Determine the validity of a logical argument.

Use valid forms of deductive reasoning, including the law of

syllogism, the law of the contrapositive, the law of detachment,

and counterexamples.

Select and use various types of reasoning and methods of proof,

as appropriate.

Use Venn diagrams to represent set relationships, such as

intersection and union.

Interpret Venn diagrams.

Recognize and use the symbols of formal logic, which include →,

↔, ~, , , and .



GEOMETRY

STANDARD G.2

The student will use the relationships between angles formed by two lines cut by a transversal to

a) determine whether two lines are parallel;

b) verify the parallelism, using algebraic and coordinate methods as well as deductive proofs; and

c) solve real-world problems involving angles formed when parallel lines are cut by a transversal.


Parallel lines intersected by a transversal form angles with


Some angle relationships may be used when proving two lines

intersected by a transversal are parallel.

The Parallel Postulate differentiates Euclidean from non-

Euclidean geometries such as spherical geometry and hyperbolic

geometry.



Use algebraic and coordinate methods as well as deductive proofs

to verify whether two lines are parallel.

Solve problems by using the relationships between pairs of angles

formed by the intersection of two parallel lines and a transversal

including corresponding angles, alternate interior angles, alternate

exterior angles, and same-side (consecutive) interior angles.

Solve real-world problems involving intersecting and parallel

lines in a plane.



GEOMETRY

STANDARD G.3

The student will use pictorial representations, including computer software, constructions, and coordinate methods, to solve problems

involving symmetry and transformation. This will include

a) investigating and using formulas for finding distance, midpoint, and slope;

b) applying slope to verify and determine whether lines are parallel or perpendicular;

c) investigating symmetry and determining whether a figure is symmetric with respect to a line or a point; and

d) determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods.


Transformations and combinations of transformations can be used

to describe movement of objects in a plane.

The distance formula is an application of the Pythagorean

Theorem.

Geometric figures can be represented in the coordinate plane.

Techniques for investigating symmetry may include paper

folding, coordinate methods, and dynamic geometry software.

Parallel lines have the same slope.

The product of the slopes of perpendicular lines is -1.

The image of an object or function graph after an isomorphic

transformation is congruent to the preimage of the object.



Find the coordinates of the midpoint of a segment, using the

midpoint formula.

Use a formula to find the slope of a line.

Compare the slopes to determine whether two lines are parallel,

perpendicular, or neither.

Determine whether a figure has point symmetry, line symmetry,

both, or neither.

Given an image and preimage, identify the transformation that has

taken place as a reflection, rotation, dilation, or translation.

Apply the distance formula to find the length of a line segment

when given the coordinates of the endpoints.



GEOMETRY

STANDARD G.4

The student will construct and justify the constructions of

a) a line segment congruent to a given line segment;

b) the perpendicular bisector of a line segment;

c) a perpendicular to a given line from a point not on the line;

d) a perpendicular to a given line at a given point on the line;

e) the bisector of a given angle;

f) an angle congruent to a given angle; and

g) a line parallel to a given line through a point not on the given line.


Construction techniques are used to solve real-world problems in

engineering, architectural design, and building construction.

Construction techniques include using a straightedge and

compass, paper folding, and dynamic geometry software.



Construct and justify the constructions of

– a line segment congruent to a given line segment;

– the perpendicular bisector of a line segment;

– a perpendicular to a given line from a point not on the line;

– a perpendicular to a given line at a point on the line;

– the bisector of a given angle;

– an angle congruent to a given angle; and

– a line parallel to a given line through a point not on the

given line.

Construct an equilateral triangle, a square, and a regular hexagon

inscribed in a circle.†

Construct the inscribed and circumscribed circles of a triangle.†

Construct a tangent line from a point outside a given circle to the

circle.†

†Revised March 2011


TOPIC: TRIANGLES

GEOMETRY

STANDARD G.5

The student, given information concerning the lengths of sides and/or measures of angles in triangles, will

a) order the sides by length, given the angle measures;

b) order the angles by degree measure, given the side lengths;

c) determine whether a triangle exists; and

d) determine the range in which the length of the third side must lie.

These concepts will be considered in the context of real-world situations.


The longest side of a triangle is opposite the largest angle of the

triangle and the shortest side is opposite the smallest angle.

In a triangle, the length of two sides and the included angle

determine the length of the side opposite the angle.

In order for a triangle to exist, the length of each side must be

within a range that is determined by the lengths of the other two

sides.



Order the sides of a triangle by their lengths when given the

measures of the angles.

Order the angles of a triangle by their measures when given the

lengths of the sides.

Given the lengths of three segments, determine whether a triangle

could be formed.

Given the lengths of two sides of a triangle, determine the range

in which the length of the third side must lie.

Solve real-world problems given information about the lengths of

sides and/or measures of angles in triangles.


TOPIC: TRIANGLES

GEOMETRY

STANDARD G.6

The student, given information in the form of a figure or statement, will prove two triangles are congruent, using algebraic and coordinate



Congruence has real-world applications in a variety of areas,

including art, architecture, and the sciences.

Congruence does not depend on the position of the triangle.

Concepts of logic can demonstrate congruence or similarity.

Congruent figures are also similar, but similar figures are not

necessarily congruent.



Use definitions, postulates, and theorems to prove triangles

congruent.

Use coordinate methods, such as the distance formula and the

slope formula, to prove two triangles are congruent.

Use algebraic methods to prove two triangles are congruent.


TOPIC: TRIANGLES

GEOMETRY

STANDARD G.7

The student, given information in the form of a figure or statement, will prove two triangles are similar, using algebraic and coordinate



Similarity has real-world applications in a variety of areas,

including art, architecture, and the sciences.

Similarity does not depend on the position of the triangle.

Congruent figures are also similar, but similar figures are not

necessarily congruent.



Use definitions, postulates, and theorems to prove triangles

similar.

Use algebraic methods to prove that triangles are similar.

Use coordinate methods, such as the distance formula, to prove

two triangles are similar.


TOPIC: TRIANGLES

GEOMETRY

STANDARD G.8

The student will solve real-world problems involving right triangles by using the Pythagorean Theorem and its converse, properties of

special right triangles, and right triangle trigonometry.


The Pythagorean Theorem is essential for solving problems

involving right triangles.

Many historical and algebraic proofs of the Pythagorean Theorem

exist.

The relationships between the sides and angles of right triangles

are useful in many applied fields.

Some practical problems can be solved by choosing an efficient

representation of the problem.

Another formula for the area of a triangle is1

sin2

A ab C .

The ratios of side lengths in similar right triangles

(adjacent/hypotenuse or opposite/hypotenuse) are independent of

the scale factor and depend only on the angle the hypotenuse

makes with the adjacent side, thus justifying the definition and

calculation of trigonometric functions using the ratios of side

lengths for similar right triangles.



Determine whether a triangle formed with three given lengths is a

right triangle.

Solve for missing lengths in geometric figures, using properties of

45-45-90 triangles.

Solve for missing lengths in geometric figures, using properties of

30-60-90 triangles.

Solve problems involving right triangles, using sine, cosine, and

tangent ratios.

Solve real-world problems, using right triangle trigonometry and

properties of right triangles.

Explain and use the relationship between the sine and cosine of

complementary angles.†



TOPIC: POLYGONS AND CIRCLES

GEOMETRY

STANDARD G.9

The student will verify characteristics of quadrilaterals and use properties of quadrilaterals to solve real-world problems.


The terms characteristics and properties can be used

interchangeably to describe quadrilaterals. The term

characteristics is used in elementary and middle school

mathematics.

Quadrilaterals have a hierarchical nature based on the

relationships between their sides, angles, and diagonals.

Characteristics of quadrilaterals can be used to identify the

quadrilateral and to find the measures of sides and angles.



Solve problems, including real-world problems, using the

properties specific to parallelograms, rectangles, rhombi, squares,

isosceles trapezoids, and trapezoids.

Prove that quadrilaterals have specific properties, using

coordinate and algebraic methods, such as the distance formula,

slope, and midpoint formula.

Prove the characteristics of quadrilaterals, using deductive

reasoning, algebraic, and coordinate methods.

Prove properties of angles for a quadrilateral inscribed in a circle.†




GEOMETRY

STANDARD G.10

The student will solve real-world problems involving angles of polygons.


A regular polygon will tessellate the plane if the measure of an

interior angle is a factor of 360.

Both regular and nonregular polygons can tessellate the plane.

Two intersecting lines form angles with specific relationships.

An exterior angle is formed by extending a side of a polygon.

The exterior angle and the corresponding interior angle form a

linear pair.

The sum of the measures of the interior angles of a convex

polygon may be found by dividing the interior of the polygon into

nonoverlapping triangles.



Solve real-world problems involving the measures of interior and

exterior angles of polygons.

Identify tessellations in art, construction, and nature.

Find the sum of the measures of the interior and exterior angles of

a convex polygon.

Find the measure of each interior and exterior angle of a regular

polygon.

Find the number of sides of a regular polygon, given the measures

of interior or exterior angles of the polygon.



GEOMETRY

STANDARD G.11 The student will use angles, arcs, chords, tangents, and secants to

a) investigate, verify, and apply properties of circles;

b) solve real-world problems involving properties of circles; and

c) find arc lengths and areas of sectors in circles.


Many relationships exist between and among angles, arcs,

secants, chords, and tangents of a circle.

All circles are similar.

A chord is part of a secant.

Real-world applications may be drawn from architecture, art, and

construction.



Find lengths, angle measures, and arc measures associated with

– two intersecting chords;

– two intersecting secants;

– an intersecting secant and tangent;

– two intersecting tangents; and

– central and inscribed angles.

Calculate the area of a sector and the length of an arc of a circle,

using proportions.

Solve real-world problems associated with circles, using

properties of angles, lines, and arcs.

Verify properties of circles, using deductive reasoning, algebraic,

and coordinate methods.



GEOMETRY

STANDARD G.12

The student, given the coordinates of the center of a circle and a point on the circle, will write the equation of the circle.


A circle is a locus of points equidistant from a given point, the

center.

Standard form for the equation of a circle is

2 2 2x h y k r , where the coordinates of the center of the

circle are ( , )h k and r is the length of the radius.

The circle is a conic section.



Identify the center, radius, and diameter of a circle from a given

standard equation.

Use the distance formula to find the radius of a circle.

Given the coordinates of the center and radius of the circle,

identify a point on the circle.

Given the equation of a circle in standard form, identify the

coordinates of the center and find the radius of the circle.

Given the coordinates of the endpoints of a diameter, find the

equation of the circle.

Given the coordinates of the center and a point on the circle, find

the equation of the circle.

Recognize that the equation of a circle of given center and radius

is derived using the Pythagorean Theorem.†



TOPIC: THREE-DIMENSIONAL FIGURES

GEOMETRY

STANDARD G.13

The student will use formulas for surface area and volume of three-dimensional objects to solve real-world problems.


The surface area of a three-dimensional object is the sum of the

areas of all its faces.

The volume of a three-dimensional object is the number of unit

cubes that would fill the object.



Find the total surface area of cylinders, prisms, pyramids, cones,

and spheres, using the appropriate formulas.

Calculate the volume of cylinders, prisms, pyramids, cones, and

spheres, using the appropriate formulas.

Solve problems, including real-world problems, involving total

surface area and volume of cylinders, prisms, pyramids, cones,

and spheres as well as combinations of three-dimensional figures.

Calculators may be used to find decimal approximations for

results.


TOPIC: THREE-DIMENSIONAL FIGURES

GEOMETRY

STANDARD G.14

The student will use similar geometric objects in two- or three-dimensions to

a) compare ratios between side lengths, perimeters, areas, and volumes;

b) determine how changes in one or more dimensions of an object affect area and/or volume of the object;

c) determine how changes in area and/or volume of an object affect one or more dimensions of the object; and

d) solve real-world problems about similar geometric objects.


A change in one dimension of an object results in predictable

changes in area and/or volume.

A constant ratio exists between corresponding lengths of sides of

similar figures.

Proportional reasoning is integral to comparing attribute measures

in similar objects.



Compare ratios between side lengths, perimeters, areas, and

volumes, given two similar figures.

Describe how changes in one or more dimensions affect other

derived measures (perimeter, area, total surface area, and volume)

of an object.

Describe how changes in one or more measures (perimeter, area,

total surface area, and volume) affect other measures of an object.

Solve real-world problems involving measured attributes of

similar objects.

hcps geometry curriculum guide

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