chapter 1 basics of geometry chapter objectives using inductive reasoning to identify patterns....

Chapter 1

Basics of Geometry

Chapter Objectives Using Inductive Reasoning to identify patterns. Identify collinear characteristics. Utilize Distance and Midpoint Formulas Label acute, obtuse, right, and straight angles. Identify angle measures Apply Angle Addition Postulate Compare Complimentary v Supplementary

angles Define angle and segment bisectors Identify the basics of perimeter, circumference,

and area

Lesson 1.1

PatternsAnd

Inductive Reasoning

Lesson 1.1 Objectives Identify patterns in numbers and shape

sequences. Use inductive reasoning. Define Geometry.

Geometry is… Geometry is a branch of mathematics

that deals with the measurement, properties, and relationships of points, lines, angles, surfaces, and solids; the study of properties of given elements that remain invariant under specified transformations.

Basically what that means is geometry is the study of the laws that govern the patterns and elements of mathematics.

Definition from Merriam-Webster Online Dictionary.

Inductive Reasoning Inductive Reasoning is the process in

which one looks for patterns in samples and makes conjectures of how the pattern will work for the entire population.

A conjecture is an unproven statement based on observations.

A conjecture is math’s version of a hypothesis, or educated guess. The education comes from the observation.

Using Inductive Reasoning Much of the reasoning in

Geometry consists of three stages1. Look for a Pattern. Look at examples

and organize any ideas of a pattern into a diagram or table.

2. Make a Conjecture. Use the examples to try to identify what step was taken to get from element to element in the pattern.

3. Verify the Conjecture. Use logical reasoning to verify the conjecture is true for all cases.

Example 1

Identify the next member of the group:

1 , 4 , 7 , 10

131 , 4 , 9 , 16

25

Counterexamples A counterexample is one example

that shows a conjecture is false. Therefore to prove a conjecture is

true, it must be true for all cases.

Conjecture: Every month has at least 30 days.

Counterexample: February has 28 (or 29).

Goldbach’s Conjecture In the early 1700s a Prussian

mathematician named Goldbach noticed that many even numbers greater than 2 can be written as the sum of two primes.

4 = 2 + 2

6 = 3 + 3

8 = 3 + 5

10 = 3 + 7

This conjecture is unproven for all cases, but has been proved for all even numbers up to 4 x 1014.400,000,000,000,000

Homework 1.1 In-class

1-11 Page 6-9

Homework 12-26 ev, 27, 28-46 ev, 47, 48, 52-70 even

Due Tomorrow

Lesson 1.2

Points, Lines, and Planes

Lesson 1.2 Objectives Define the basic terms of geometry. Sketch the basic components of

geometrical figures.

Start-Up Give your

definition of the following Point Line Plane

Not an airplane!

These terms are actually said to be undefined, or have no formal definition.

However, it is important to have a general agreement on what each word means.

Point A point has no dimension.

Meaning it takes up no space. It is usually represented as a dot. When labeling we designate a

capital letter as a name for that point. We may call it Point A. A

Line A line extends in one dimension.

Meaning it goes straight in either a vertical, horizontal, or slanted fashion.

It extends forever in two directions. It is represented by a line with an arrow on

each end. When labeling, we use lower-case letters to

name the line. Or the line can be named using two points that are

on the line. So we say Line n, or AB

A

Bn

Plane A plane extends in two dimensions.

Meaning it stretches in a vertical direction as well as a horizontal direction at the same time.

It also extends forever. It is usually represented by a shape like a

tabletop or a wall. When labeling we use a bold face capital letter

to name the plane. Plane M Or the plane can be named by picking three points in

the plane and saying Plane ABC.

A

B

C

M

Collinear The prefix co- means the same, or

to share. Linear meansline.

A B C

We say that points A, B, and C are collinear.

So collinear means that points lie on the same line.

Coplanar Coplanar points are points that

lie on the same plane.

A

B

C

M

So points A, B, and C are said to be coplanar.

Line Segment Consider the line AB.

It can be broken into smaller pieces by merely chopping the arrows off.

This creates a line segment or segment that consists of endpoints A and B. This is symbolized as

A

B

AB

Ray A ray consists of an initial point

where the figure begins and then continues in one direction forever. It looks like an arrow.

This is symbolized by writing its initial point first and then naming any other point on the ray, . Or we can say ray AB.

AB

A

B

Opposite Rays If C is between A and B on a line,

then ray CA and ray CB are opposite rays. Opposite rays are only opposite if

they are collinear.

A BC

Intersections ofLines and Planes Two or more geometric figures intersect if

they have one or more points in common. If there is no point or points shown, they the figures

do not intersect. The intersection of the figures is the set of points

the figures have in common. Two lines intersect at one point. Two planes intersect at one line.

A

m n

Homework 1.2 In-class

1-8 Page 13-15

Homework 9-42 every 3rd, 44 – 51, 56-66 ev, 68-75

Due Tomorrow

Lesson 1.3

Segments and Their Measures

Lesson 1.3 Objectives Define what a postulate is. Use segment postulates. Utilize the Distance Formula. Identify congruent segments.

Definition of a Postulate A postulate is a rule that is

accepted without a proof. They may also be called an axiom.

Basically we do not need to know the reason for the rule when it is a postulate.

Postulates are used together to prove other rules that we call theorems.

Postulate 1: Ruler Postulate The points on a line can be

matched to real numbers called coordinates.

The distance between the points, say A and B, is the absolute value of the difference of the coordinates. Distance always positive.

A B

4 8

Length Finding the distance between

points A and B is written as AB

Writing AB is also called the length of line segment AB.

Betweenness When three points lie on a line, we

can say that one of them is between the other two. This is only true if all three points are

collinear. We would say that B is between A and

C.

A B C

Postulate 2: Segment Addition Postulate

If B is between A and C, then AB + BC = AC.

Also, the opposite is true. If AB + BC = AC, then B is between A

and C.

A B C

AB BC

AC

Lesson 1.3A Homework In-class

1, 3, 4-8 ev, 11 p21-22

Homework 13-33

Due Tomorrow Quiz Monday

Lessons 1.1-1.3

Lesson 1.3

Part II

Segment Addition Postulate Review Identify the unknown lengths given that BD=4, AE=17, AD=7, and BC=CD

BC 2

AB 3

AC 5

DE 10

A B C D E

Distance FormulaTo find the distance on a graph between two points

A(1,2) B(7,10) We use the Distance Formula

(x2 – x1)2 + (y2 – y1)2AB =

Distance can also be found using the Segment Addition Postulate, which simply adds up each segment of a line to find the total length of the line.

Example 2 Using the Distance Formula, find the

length of segment OK with endpoints

O(2,6) K(5,10)

(x2 – x1)2 + (y2 – y1)2

(5 – 2)2 + (10 – 6)2

32 + 42

9 + 16

25 = 5

Example 3 This is one part of the problem for

#34 Find the distance between points A

and C.

A(-4,7) C(3,-2)

(x2 – x1)2 + (y2 – y1)2

(3 – -4)2 + (-2 – 7)2

72 + (-9)2

49 + 81

130

Congruent Segments Segments that have the same length

are called congruent segments. This is symbolized by =.

If you want to state two segments are congruent, then you write

LE = NT

If you want to state two lengths are equal, then you write

LE = NT

Hint: If the symbols are there, the congruent sign should be there.

Lesson 1.3B Homework In-Class

4-8 ev p22-24

Homework 37-56, 60-70 ev

skip 44,47,52-54

Due Tomorrow Quiz Tomorrow

Lesson 1.1-1.3

Lesson 1.4

Angles and Their Measures

Lesson 1.4 Objectives Use the angle postulates. Identify the proper name for

angles. Classify angles as right, obtuse,

acute, or straight. Measure the size of an angle.

What is an Angle? An angle consists of two different

rays that have the same initial point. The rays form the sides of the angle. The initial point is called the vertex

of the angle. Vertex can often be thought of as a

corner.

Naming an Angle All angles are named by using three

points Name a point that lies on one side of the

angle. Name the vertex next.

The vertex is always named in the middle. Name a point that lies on the opposite side

of the angle. W

O N

So we can call It WONOr NOW

Congruent Angles Congruent angles are angles that

have the same measure. To show that we are finding the measure of

an angle Place a “m” before the name of the angle.

WONm NOWm=

Equal Measures

WON NOW=

Congruent Angles

Types of Angles

Acute Right Obtuse Straight

Looks like

Measure

<90 =90 >90 =180

Other Parts of an Angle The interior of an angle is defined as

the set of points that lie between the sides of the angle.

The exterior of an angle is the set of points that lie outside of the sides of the angle.

Interior

Exterior

Postulate 4: Angle Addition Postulate

The Angle Addition Postulate allows us to add each smaller angle together to find the measure of a larger angle.

32o

17o

What is the total?

49o

Adjacent Angles Two angles are adjacent angles if

they share a common vertex and side, but have no common interior points. Basically they should be touching, but

not overlapping. C

A

T

R

CAT and TAR are adjacent.

CAR and TARare not adjacent.

Homework Lesson 1.4 In-Class

1-16 p29-32

Homework 18-48 ev, 50-53, 62-78 ev

skip 30,32,34

Due Tomorrow

Lesson 1.5

Segmentsand

Angle Bisectors

Lesson 1.5 Objectives Identify a segment bisector Identify an angle bisector Utilize the Midpoint Formula

Midpoint The midpoint of a segment is the point that

divides the segment into two congruent segments. The midpoint bisects the segment, because bisect means

to divide into two equal parts. A segment bisector is a segment, ray, line, or

plane that intersects the original segment at its midpoint.

J YOWe say that O is the midpointof line segment JY.

Congruence marks are used to show thatsegments are congruent. If there is more than one pair of congruent segments, then each pair should get a different number of congruence marks.

H

TNow we can sayline HT is asegment bisector of segment JY.

Midpoint Formula

A(1,2) B(7,10)

We can also find the midpoint of segment AB byusing its endpoints in…

The Midpoint Formula

Midpoint of AB = (x1 + x2)

(y1 + y2)( ),2 2

This gives the coordinates of the midpoint, or point that is halfway between A and B.

Example 4 This is an example of how to determine the

midpoint knowing the two endpoints.

A(1,2) B(7,10)(x1 + x2) (y1 + y2)( ),2 2

(1 + 7) (2 + 10)( ),2 2

8 12( ),2 2

( ),4 6

Example 5 This is an example of how to find an endpoint

knowing the midpoint and the other endpoint. Say the midpoint is (8,5) and one endpoint is (4,9).

( ),(x1 + x2) (y1 + y2)

2 2

Remember that each coordinate from the midpoint was found from…

So, use each coordinate from the midpoint formula to solve for x2.

(x1 + x2)

2= 8

(4 + x2)

2= x 2

x 2

4 + x2=16-4-4

x2= 12

(y1 + y2)

2= 5

(9 + y2)

2= x 2

x 2

9 + x2=10-9-9

y2= 1

So the coordinates for the other endpoint are

(12,1)

Short Cut to Find Endpoint Say the midpoint is (8,5) and one endpoint is (4,9). Remember that the midpoint is half way between

the endpoints.

(4,9)

(8,5)

Add 4 to x

Minus 4 from y

Add 4 to x

Minus 4 from y

(12,1)

Angle Bisector An angle bisector is a ray that

divides an angle into two adjacent angles that are congruent. To show that angles are congruent,

we use congruence arcs.

Homework 1.5 In-Class

1-2, 4-13 p38-42

Homework 18-32 ev, 38-54, 62-72 ev

Due Tomorrow

Lesson 1.6

Angle Pair Relationships

Lesson 1.6 Objectives Identify vertical angle pairs. Identify linear pairs. Differentiate between

complementary and supplementary angles.

Vertical Angles Two angles are vertical angles if their sides

form two pairs of opposite rays. Basically the two lines that form the angles are

straight. To identify the vertical angles, simply look

straight across the intersection to find the angle pair.

Hint: The angle pairs do not have to be vertical in position.

Vertical Angle pairs are always congruent!1

324

1

324

Linear Pair Two adjacent angles form a linear pair if

their non-common sides are opposite rays. Simply put, these are two angles that share a

straight line. Since they share a straight line, their sum

is… 180o

1 2

Complementary v Supplementary Complementary

angles are two angles whose sum is 90o. Complementary

angles can be adjacent or non-adjacent.

Supplementary angles are two angles whose sum is 180o. Supplementary

angles can be adjacent or non-adjacent.

Homework 1.6 In-Class

1, 4-7 p47-50

Homework 8-40 ev, 46-54 ev, 60-74 ev

Due Tomorrow Test Tuesday

Lesson 1.7

Intro to Perimeter,Circumference and

Area

Lesson 1.7 Objectives Find the perimeter and area of

common plane figures. Establish a plan for problem

solving.

Perimeter and Area of a Rectangle Recall that the

perimeter of a figure is the sum of the lengths of the sides.

A rectangle has two pairs of opposite sides that are congruent. l + w + l + w =

P = 2l + 2w

Recall the area of a figure is the measure of space inside the figure.

This is found by taking the length of the rectangle times the width of the rectangle. l • w =

A = lw

Perimeter and Area of a Square Since a square has

four congruent sides, the formula is quite simple…

s + s + s + s = P = 4s

Since a square is also a rectangle, we can find the area by multiplying the length times the width

s • s = A = s2

s

Perimeter and Area of a Triangle The perimeter can be

found by adding the three sides together.

P = a + b + c If the third side is

unknown, use the Pythagorean Theorem to solve for the unknown side.

a2 + b2 = c2

Where a,b are the two shortest sides and c is the longest side.

The area of a triangle is half the length of the base times the height of the triangle.

The height of a triangle is the perpendicular length from the base to the opposite vertex of the triangle.

A = ½bh

ab

c

h

Circumference and Area of a Circle

The perimeter of a circle is called the circumference.

It is found by taking the diameter times .

C = d or 2r r is the radius, which

is half the diameter.

The area is found by taking pi times the radius squared.

A = r2

d=diameter

r=radius

Homework In-Class

1-8 p55-58

Homework 10-32 ev, 38-48 ev, 58-62 ev

Due Tomorrow Test Tuesday

December 18th