1.1: Points, Lines, and Planes

Undefined Term - can only be explained using examples and descriptions Point – a single spot in space with 0 dimensions Line – extends infinitely in opposite directions and is 1 dimensional Plane – extends infinitely in 4 directions and is 2 dimensional Collinear Points – points that lie on the same line Coplanar Points – points that lie on the same plane Coplanar Lines – lines that lie on the same plane Space - boundless, 3-dimensional set of all points

1.2: Linear Measure and Precision

Line Segment – has endpoints Endpoint – point at one end of a segment or the starting point for a ray Precision - depends on the smallest unit available on the measuring tool

(Accuracy) Congruent Segments – segments with the same length Construction – a way of creating a precise figure without measuring tools

1.3: Distance and Midpoints

Coordinate – real number corresponding to a point Distance/Length – absolute value of the difference between coordinates Distance Formula - √(X2 – X1)2 + (Y2 – Y1)2

Midpoint – the point that bisects or divides a segment into two congruent segments

Midpoint Formula – (X1 + X2)/2 , (Y1 + Y2)/2 Coordinate Plane – plane that is divided into four quadrants by the x and y axis Bisect – to divide equally Segment Bisector – any ray, segment, or line that intersects a segment at its

midpoint

1.4: Angle Measures

Degree – 1/360th of a circle

Ray – has one endpoint and extends infinitely in the other direction Opposite Rays – have a common endpoint, going in opposite directions, and form

a line Angle – consists of two rays with a common endpoint Sides of the Angle – are the rays Vertex of the Angle – is the common endpoint Interior of the Angle – the set of all points between the sides of the angle Exterior of the Angle – the set of all points outside the sides of the angle Measure of an Angle – measured in degrees 3 Pieces of an Angle – Interior, Exterior, and the Angle itself

Acute Angle – less than 90 degrees Obtuse Angle – greater than 90 degrees Right Angle – equals 90 degrees Straight Angle – equals 180 degrees Congruent Angles – have the same measure Angle Bisector – a ray that divides an angle into two congruent angles Angle Addition Postulate – if P is in the interior of angle RST then the measure of

angle RSP plus the measure of angle PST = measure of angle RST

1.5: Angle Relationships

Adjacent Angles – two angles in the same plane with a common vertex and side but no common interior points

Vertical Angles – two non-adjacent angles formed by two intersecting lines Linear Pair – a pair of adjacent angles whose non-common sides are opposite rays Complementary Angles – two angles whose measures have a sum of 90 degrees Supplementary Angles – two angles whose measures have a sum of 180 degrees Perpendicular - to form right angles

1.6: 2-Dimensional Figures

Polygon – a plane figure formed by 3 or more segments, each intersecting with exactly two other segments at their endpoints

Sides – the segments of a polygon Vertex – formed by the intersection of two sides at an endpoint Diagonal – a segment connecting two non-consecutive vertices Triangle – 3 sides Quadrilateral – 4 sides Pentagon – 5 sides Hexagon – 6 sides Heptagon – 7 sides Octagon – 8 sides Nonagon – 9 sides Decagon – 10 sides Dodecagon – 12 sides To name any polygon just add “Gon” to the number of sides, example; 13-gon Convex – no line that contains a side passes through the interior of the polygon Concave – a line containing a side passes through the interior of the polygon Regular Polygon - all the sides and angles are congruent Perimeter – the distance around a 2 dimensional figure Area – amount of space a 2 dimensional figure occupies measured in square units Perimeter of a Rectangle – 2(L + W) Area of a Rectangle – LW Perimeter of a Square – 4S Area of a Square – S2

Perimeter of a Triangle – A + B + C

Area of a Triangle – 1/2BH Diameter – segment that passes through the center of a circle Radius – length from the center to the circle itself Circumference – distance around a circle Circumference of a Circle - 2πR or πD Area of a Circle – πR2

1.7: 3-Dimensional Figures

Solids – 3 dimensional figures Polyhedron – solid formed using polygons To name a polyhedron use the name of the shape of its base Base – two congruent polygons in parallel planes Faces – sides of the polyhedron Edges – segments formed by the intersection of two polygons Vertices – point where 3 or more faces meet Prism – type of polyhedron formed by two parallel congruent polygon bases with

the faces being parallelograms Regular Prism - the bases are regular polygons Pyramid – type of polyhedron formed by a polygon base with the faces being

triangles that meet at a common vertex Regular Polyhedron – all faces are congruent regular polygons Cross Section – intersection of a 3-dimensional figure with a plane Platonic Solids – the 5 regular polyhedron are tetrahedron (4 faces), cube (6

faces), octahedron (8 faces), dodecahedron (12 faces), icosahedron (20 faces) Cylinder – solid formed by two parallel congruent circles as bases with a curved

surface connecting the bases Cone – solid formed using one circle as a base with a curved surface connecting

the base to a vertex Sphere – a solid made up of a set of points all a given distance from the center' Surface Area - sum of the areas of each face of the solid, measured in square units Volume - the amount of space inside a solid, measured in cubic units Surface Area of a Right Prism – 2B + PH, where B is the area of the base, P is

the perimeter around the base, and H is the height between the bases Volume of a Prism – BH, where B is the area of the base, and H is the

perpendicular height between the bases Surface Area of a Right Cylinder – 2B + CH, where B is the area of the base, C is

the circumference of the base, and H is the height between the bases Volume of a Cylinder – BH, where B is the area of the base, and H is the

perpendicular height between the bases Surface Area of a Right Pyramid – B + 1/2Pℓ, where B is the area of the base, P is

the perimeter of the base, and ℓ is the slant height of a lateral face Volume of a Pyramid – 1/3BH, where B is the area of the base, and H is the

perpendicular height from the base to the opposite vertex Surface Area of a Right Cone – B + πRℓ, where B is the area of the base, R is the

radius of the base, and ℓ is the slant height of the lateral face

Volume of a Cone – 1/3πR2H, where R is the radius of the base, and H is the perpendicular height from the base to the opposite vertex

Surface Area of a Sphere - 4πR2, where R is the radius Volume of a Sphere – 4/3πR3, where R is the radius

2.1: Inductive Reasoning and Conjectures

Conjecture – a statement believed to be true based on inductive reasoning Inductive Reasoning – the process of making true statements based on patterns Counterexample – an example that nullifies a conjecture

2.2: Logic

Statement - is any sentence that is true or false, but not both Truth Value - either true or false Negation - opposite meaning of the truth value Compound Statement – is created by combining two or more statements Conjunction – compound statement using the word “And” Disjunction – compound statement using the work “Or”

2.3: Conditional Statements

Conditional Statement – a statement that can be written as “If p, then q” Hypothesis – follows the word “If” in the conditional statement Conclusion – follows the word “Then” in the conditional statement Truth Value – either true or false Related Conditionals - other statements based on the original conditional Converse – flips the hypothesis and conclusion of a conditional statement Inverse – negation of the hypothesis and conclusion of a conditional statement Contrapositive – flips the hypothesis and conclusion of the inverse Logically Equivalent Statements – have the same truth value, example; the

conditional and contrapositive, and the converse and inverse Biconditional Statement – is a statement that can be written in the form “P if and

only if Q” meaning both “If p, then q” and “If q, then p” Definition – is a statement that describes a mathematical object and can be written

as a true biconditional

2.4: Deductive Reasoning

Deductive Reasoning – process of making true statements based on given facts, definitions, and properties

Law of Detachment – if the hypothesis of a true if-then statement is true, then the conclusion must also be true

Law of Syllogism – if p then q, and if q then r are both true statements then if p then r must also be true

2.5: Postulates and Paragraph Proofs

Postulate (Axiom) – statement accepted without further justification Postulate 2.1 – through any two points there is exactly one line Postulate 2.2 – through any three non-collinear points there is exactly one plane Postulate 2.3 - a line contains at least 2 points Postulate 2.4 - a plane contains at least 3 non-collinear points Postulate 2.5 – if two points lie in a plane, then the line containing those points

lies in the plane Postulate 2.6 – two lines intersect at a point Postulate 2.7 – two planes intersect in a line Theorem - a statement proven to be true Proof - an argument that uses logic, definitions, properties, and previously proven

statements to show that a conclusion is true Equal – refers to measurement Congruence – refers to the physical size and shape of a figure Theorem 2.1 - If M is the midpoint of AB, then AM is congruent to MB

2.6: Algebraic Proof

Reflexive Property – for any real number A, A=A Symmetric Property – If A = B, then B = A Transitive Property – If A = B, and B = C, then A = C Addition Property – If A = B, then A + C = B + C Subtraction Property – If A = B, then A – C = B – C Multiplication Property – If A = B, then AC = BC Division Property – If A = B, then A/C = B/C Substitution Property – If A = B, then A can be substituted for B in any equation

or expression Three Properties of Congruence – reflexive, symmetric, and transitive and are the

same as equality only referenced towards congruence 2 Column Proof – a proof organized of 2 columns, one with statements the other

with reasons

2.7: Proving Segment Relationships

Postulate 2.8 - the points on any line or segment can be paired with real numbers on a number line

Postulate 2.9 - if A, B, and C are collinear, and B is between A and C, then AB + BC = AC

Theorem 2.2 - Congruence of segments is reflexive, symmetric, and transitive

2.8: Proving Angle Relationships

Postulate 2.11 - If R is in the interior of angle PQS, then angle PQR + angle RQS = angle PQS

Theorem 2.3 - if two angles form a linear pair then they are supplementary Theorem 2.4 - if the non-common sides of 2 adjacent angles form a right angle,

then the angles are complementary Theorem 2.5 - Congruence of angles is reflexive, symmetric, and transitive Theorem 2.6 - Angles supplementary to the same or congruent angles are

congruent Theorem 2.7 - Angles complementary to the same or congruent angles are

congruent Theorem 2.8 - Vertical angles are congruent Theorem 2.9 - Perpendicular lines form 4 right angles Theorem 2.10 - All right angles are congruent Theorem 2.11 - Perpendicular lines form congruent adjacent angles Theorem 2.12 - If 2 angles are congruent and supplementary then they are right

angles Theorem 2.13 - If 2 congruent angles form a linear pair then they are right angles

3.1: Parallel Lines and Transversals

Parallel Lines – coplanar lines that do not intersect Parallel Planes – do not intersect Skew Lines – non-coplanar lines that do not intersect Perpendicular Lines – intersect to form right angles Line Perpendicular to a Plane – intersects a plane at a point and is perpendicular

to every line in the plane that it intersects Transversal – intersects two or more coplanar lines at different points Corresponding Angles – occupy corresponding positions Alternate Interior Angles – lie between the coplanar lines on opposite sides of the

transversal Alternate Exterior Angles – lie outside the coplanar lines on opposite sides of the

transversal Consecutive/Same-Side Interior Angles – lie between the coplanar lines on the

same side of the transversal

3.2: Angles and Parallel Lines

Postulate 3.1 (Corresponding Angles Postulate) – if two parallel lines are cut by a transversal, then the corresponding angles are congruent

Theorem 3.1 (Alternate Interior Angles Theorem) – if two parallel lines are cut by a transversal, then the alternate interior angles are congruent

Theorem 3.2 (Alternate Exterior Angles Theorem) – if two parallel lines are cut by a transversal, then the alternate exterior angles are congruent

Theorem 3.3 (Consecutive/Same-Side Interior Angles Theorem) – if two parallel lines are cut by a transversal, then the consecutive/same-side interior angles are supplementary

Theorem 3.4 - In a plane, if a line is perpendicular to one of two parallel lines, then it perpendicular to the other

3.3: Slopes of Lines

Slope – the ratio of the change in Y values to the change in X values of a line Y2 - Y1

X2 - X1

Rise – the difference in Y values of two points on a line Run – the difference in X values of two points on a line Rate of Change - describes how a quantity changes over time Parallel Lines Theorem – in a coordinate plane, two non-vertical lines are parallel

if and only if they have the same slope, and vertical lines are always parallel Perpendicular Lines Theorem – in a coordinate plane, two non-vertical lines are

perpendicular if and only if the product of their slopes is -1, vertical and horizontal lines are always perpendicular

3.4: Lines in the Coordinate Plane

Slope-Intercept Form – Y = MX + B Point-Slope Form – Y – Y1 = M(X – X1) Parallel Lines – same slope different y-intercept Intersecting Lines – different slopes Coinciding Lines – same slope and y-intercept

3.5: Proving Lines are Parallel

Postulate 3.4 (Corresponding Angles Converse) – if corresponding angles are congruent, then the lines cut by the transversal are parallel

Postulate 3.5 (Parallel Postulate) – if there is a line and a point not on the line, then there is exactly one line that can be drawn through the point that is parallel to the line

Theorem 3.5 (Consecutive/Same-Side Interior Angles Converse) – if consecutive/same-side interior angles are supplementary, then the lines cut by the transversal are parallel

Theorem 3.6 (Alternate Interior Angles Converse) – if alternate interior angles are congruent, then the lines cut by the transversal are parallel

Theorem 3.7 (Alternate Exterior Angles Converse) – if alternate exterior angles are congruent, then the lines cut by the transversal are parallel

Theorem 3.8 - In a plane, if 2 lines are perpendicular to the same line, then they are parallel

3.6: Perpendiculars and Distance

Distance from a Point to a Line – is always a perpendicular distance Equidistant - same distance

Theorem 3.9 – if two coplanar lines are perpendicular to the same line, then they are parallel to each other

Perpendicular Bisector – a line perpendicular to a segment at its midpoint

4.1: Classifying Triangles

Triangle – 3 sided figure formed by 3 segments connecting three non-collinear points

Acute Triangle – all three angles are acute Right Triangle – one angle equals 90 degrees, the other two are acute Obtuse Triangle – one angle is obtuse, the other two are acute Equiangular Triangle – all three angles are congruent Equilateral Triangle – all three sides congruent Isosceles Triangle – at least 2 sides congruent Scalene Triangle – no sides congruent Vertex – point connecting two sides of a triangle

4.2: Angles of Triangles

Theorem 4.1 (Triangle Sum Theorem) – sum of the measures of the interior angles of a triangle is 180 degrees

Theorem 4.2 (Third Angles Theorem) – if two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent

Interior – all set of points inside a figure Exterior – all set of points outside a figure Interior Angles – angles inside the figure, formed by two sides Exterior Angles – angles outside the figure, formed by one side and the extension

of an adjacent side Remote Interior Angle – an interior angle that is not adjacent to the exterior angle

of interest Theorem 4.3 (Exterior Angle Theorem) – the measure of an exterior angle of a

triangle is equal to the sum of the measures of the two non-adjacent interior angles

Flow Proof - organizes a series of statements in logical order Auxiliary Line – line added to a figure to aid in a proof Corollary – a statement that can be proven easily using the theorem it is related to Corollary 4.1 (Corollary to the Triangle Sum Theorem) – the acute angles of a

right triangle are complementary Corollary 4.2.- there can only be one right or obtuse angle in a triangle

4.3: Congruent Triangles

Congruent – same size and shape Corresponding Parts of Congruent Triangles are Congruent – CPCTC Theorem 4.4 - congruence of triangles is reflexive, symmetric, and transitive

Congruence Transformations - if you slide, flip, or turn a triangle, the size and shape do not change

Corresponding Parts – pieces that reside in the same position of different figures Corresponding Angles – angles in the same position of different figures Corresponding Sides – sides in the same position of different figures Congruent Polygons – if and only if their corresponding sides and angles are

congruent

4.4: Proving Congruence (SSS and SAS)

Triangle Rigidity – if the side lengths are given, then the triangle can have only one shape

Included Angle – angle that is formed by two adjacent sides Postulate 4.1 (Side-Side-Side Congruence Postulate) – if three sides of one

triangle are congruent to three sides of another triangle, then the triangles are congruent (SSS)

Postulate 4.2 (Side-Angle-Side Congruence Postulate) – if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent (SAS)

4.5: Proving Congruence (ASA, AAS, HL)

Included Side – common side between two consecutive angles Postulate 4.3 (Angle-Side-Angle Congruence Postulate) – if two angles and the

included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent (ASA)

Theorem 4.5 (Angle-Angle-Side Congruence Theorem) – if two angles and a side of one triangle are congruent to two angles and the corresponding side of a another triangle, then the triangles are congruent (AAS)

Postulate 4.4 (Hypotenuse-Leg Congruence Postulate) – if the hypotenuse and leg of one triangle are congruent to the hypotenuse and corresponding leg of a another triangle, then the triangles are congruent (HL)

4.6: Isosceles Triangles

Legs – congruent sides of an isosceles triangle Vertex Angle – angle formed by the legs Base – third non-congruent side of an isosceles triangle Base Angles – angles formed using the base as a side of the angle Theorem 4.9 (Base Angles Theorem) – if two sides of a triangle are congruent

then the angles opposite those sides are also congruent Theorem 4.10 (Converse of the Base Angles Theorem) – if two angles of a

triangle are congruent then the sides opposite those angles are also congruent Corollary 4.3 – if a triangle is equilateral, then it is also equiangular Corollary 4.4 – each angle of an equilateral triangle equals 60 degrees

4.7: Triangles and Coordinate Proof

Coordinate Proof - uses a coordinate plane and algebra to prove geometric concepts

5.1: Bisectors, Medians, and Altitudes

Equidistant – the same distance from two or more objects Perpendicular Distance – the shortest distance from a point to a line Locus – a set of points that satisfies a given condition Theorem 5.1 (Perpendicular Bisector Theorem) – if a point is on the

perpendicular bisector of a segment, then it is equidistant from the segment’s endpoints

Theorem 5.2 (Converse of the Perpendicular Bisector Theorem) – if a point is equidistant from the endpoints of a segment, then it lies on the perpendicular bisector of the segment

Concurrent – when three or more lines intersect at one point Point of Concurrency – the point the lines intersect at Circumscribed – to draw one figure around another figure Inscribed – to draw one figure inside another figure Circumcenter – point of concurrency for perpendicular bisectors, this can be

inside, on, or outside the circle depending on the triangle Theorem 5.3 (Circumcenter Theorem) – the circumcenter is equidistant from the

vertices of the triangle Theorem 5.4 (Angle Bisector Theorem) – if a point is on the bisector of an angle,

then it is equidistant from the two sides of the angle Theorem 5.5 (Converse of the Angle Bisector Theorem) – if a point is equidistant

from the sides of an angle, then it lies on the bisector of the angle Incenter – the point of concurrency for angle bisectors Theorem 5.6 (Incenter Theorem) – the incenter is equidistant from the sides of the

triangle Median of a Triangle – a segment that goes from a vertex to the midpoint of the

opposite side Centroid – point of concurrency for medians Theorem 5.7 (Centroid Theorem) – the distance from a vertex to the centroid is

two-thirds the length of that median Altitude – is a segment that goes from a vertex to the opposite side and is

perpendicular to that side Orthocenter – point of concurrency for altitudes

5.2: Inequalities and Triangles

Theorem 5.8 - an exterior angle of a triangle is greater than either of the two remote interior angles

Theorem 5.9 – the largest angle of a triangle is opposite the largest side

Theorem 5.10 – the largest side of a triangle is opposite the largest angle

5.3: Indirect Proof

Indirect Reasoning - to assume the conclusion is false and then show that this assumption leads to a contradiction of the hypothesis or some other fact

Indirect Proof – you begin by assuming the conclusion is false Writing an Indirect Proof – identify the conjecture to be proven, assume the

negation of the conclusion is true, use direct reasoning to show that the assumption leads to a contradiction, conclude that the assumption being false makes the original conjecture true

5.4: The Triangle Inequality

Theorem 5.11 – the sum of any two side lengths of a triangle must be larger than the third

Theorem 5.12 - the shortest distance from a point to a line is a perpendicular segment

Corollary 5.1 - the shortest distance from a point to a plane is a perpendicular segment

5.5: Inequalities involving Two Triangles

Theorem 5.13 (Hinge Theorem) – if two sides of one triangle are congruent to two sides of another triangle and the included angles are not congruent, then the larger third side will be opposite the larger included angle

Theorem 5.14 (Converse of the Hinge Theorem) – if two sides of one triangle are congruent to two sides of another triangle and the third sides are not congruent, then the larger included angle is opposite the larger third side

6.1: Angles of Polygons

Polygon – a plane figure formed by 3 or more segments, each intersecting with exactly two other segments at their endpoints

Sides – the segments of a polygon Vertex – formed by the intersection of two sides at an endpoint Diagonal – a segment connecting two non-consecutive vertices Triangle – 3 sides Quadrilateral – 4 sides Pentagon – 5 sides Hexagon – 6 sides Heptagon – 7 sides Octagon – 8 sides Nonagon – 9 sides Decagon – 10 sides Dodecagon – 12 sides

To name any polygon just add “Gon” to the number of sides, example; 13-gon Convex – no line that contains a side passes through the interior of the polygon Concave – a line containing a side passes through the interior of the polygon Theorem 6.1 (Polygon Angle Sum Theorem) – the sum of the interior angles of a

convex polygon is (N – 2) * 180 Individual Interior Angle – for a regular polygon, an individual interior angle is

[(N – 2)*180]/N Theorem 6.2 (Polygon Exterior Angle Sum Theorem) – the sum of the measures

of the exterior angles of a convex polygon is 360 degrees Individual Exterior Angle – for a regular polygon, an individual exterior angle is

360/N

6.2: Parallelograms

Parallelogram – a quadrilateral with both pairs of opposite sides parallel Theorem 6.3 – if a quadrilateral is a parallelogram then its opposite sides are

congruent Theorem 6.4 – if a quadrilateral is a parallelogram then its opposite angles are

congruent Theorem 6.5 – if a quadrilateral is a parallelogram then its consecutive angles are

supplementary Theorem 6.6 - if a parallelogram has 1 right angle, then all the angles are right

angles Theorem 6.7 – if a quadrilateral is a parallelogram then its diagonals bisect each

other Theorem 6.8 - each diagonal of a parallelogram splits it into 2 congruent triangles

6.3: Tests for Parallelograms

Theorem 6.9 – if both pairs of opposite sides of a quadrilateral are congruent, then it is a parallelogram

Theorem 6.10 – if both pairs of opposite angles of a quadrilateral are congruent, then it is a parallelogram

Theorem 6.11 – if the diagonals of a quadrilateral bisect each other, then it is a parallelogram

Theorem 6.12 – if one pair of opposite sides of a quadrilateral are congruent and parallel, then it is a parallelogram

Quadrilateral Angles Theorem – if an angle of a quadrilateral is supplementary to both its consecutive angles, then it is a parallelogram

6.4: Rectangles

Rectangle – parallelogram with 4 right angles Theorem 6.13 – the diagonals of a rectangle are congruent Theorem 6.14 – if the diagonals of a parallelogram are congruent, then it is a

rectangle

6.5: Rhombi and Squares Rhombus – parallelogram with 4 congruent sides Theorem 6.15 – the diagonals of a rhombus are perpendicular Rhombus Diagonals Theorem – if the diagonals of a parallelogram are

perpendicular, then it is a rhombus Theorem 6.17 – the diagonals of a rhombus bisect the opposite angles Square – parallelogram with 4 congruent sides and 4 right angles and shares all

the properties of rectangles and rhombuses

6.6: Properties of Trapezoids

Trapezoid – a quadrilateral with one pair of parallel sides Bases – the parallel sides Legs – the non-parallel sides Base Angles – angles formed by the bases Isosceles Trapezoid – the legs are congruent Theorem 6.18 – if a trapezoid is isosceles, then each pair of base angles is

congruent Theorem 6.19 – a trapezoid is isosceles if an only if the diagonals are congruent Midsegment – connects the midpoints of the legs in a trapezoid Theorem 6.20 (Midsegment Theorem) – Midsegment = (Base 1 + Base 2)/2 and it

is parallel to the bases

6.7: Properties of Kites

Kite – a quadrilateral with 2 pairs of consecutive congruent sides Theorem 6.21 – if a quadrilateral is a kite, then the diagonals are perpendicular Theorem 6.22 – if a quadrilateral is a kite, then exactly one pair of opposite angles

is congruent

7.1: Proportions

Ratio – the comparison of two numbers using division, where the denominator cannot be zero

Proportion – an equation setting two ratios equal Means – bottom left and top right in a proportion Extremes – top left and bottom right in a proportion Cross Product Property – in a proportion, the product of the extremes is equal to

the product of the means

7.2: Similar Polygons

Similar – same shape but different size Similar Polygons – corresponding angles are congruent and the corresponding

sides are proportional

Scale Factor/Similarity Ratio – ratio of the lengths of corresponding sides in similar figures

7.3: Similar Triangles

Similarity Statement – how to state two figures are similar to each other Statement of Proportionality – how to state the corresponding parts of two figures

are proportional to each other Postulate 7.1 (Angle-Angle Similarity Postulate) – if two angles of one triangle

are congruent to two angles of a another triangle, then they are similar (AA) Theorem 7.1 (Side-Side-Side Similarity Theorem) – if the sides of one triangle

are proportional to the corresponding sides of a second triangle, then they are similar (SSS)

Theorem 7.2 (Side-Angle-Side Similarity Theorem) – if two sides of one triangle are proportional to the corresponding sides of a second triangle, and the included angles are congruent, then they are similar (SAS)

Theorem 7.3 - Similarity of triangles is reflexive, symmetric, and transitive

7.4: Parallel Lines and Proportional Parts

Theorem 7.4 (Triangle Proportionality Theorem) – if a line parallel to one side of a triangle intersects the other two sides, then it divides those two sides proportionally

Theorem 7.5 (Converse of the Triangle Proportionality Theorem) – if a line divides two sides of a triangle proportionally, then it is parallel to the third side

Midsegment - connects the midpoints of two sides of a triangle Theorem 7.6 - the midsegment of a triangle is parallel to the third side and half its

length Corollary 7.1 (Two-Transversal Proportionality) – if three or more parallel lines

intersect two transversals, then they divide the transversals proportionally Corollary 7.2 - if three or more parallel lines cut congruent segments on one

transversal, then they cut congruent segments on every transversal

7.5: Parts of Similar Triangles

Indirect Measurements – any method using formulas, similar figures, and or proportions to measure an object

Scale Drawing – represents an object as smaller or larger than its actual size Scale – ratio of any length in a drawing to the corresponding actual length Theorem 7.7 (Perimeters of Similar Figures) – if two figures are similar, then the

ratio of their perimeters is equal to the ratio of the corresponding sides, which is also equal to the scale factor

Proportional Perimeters and Areas Theorem - if the ratio of the perimeters is A/B, then the ratio of the areas is A2/B2

Theorem 7.8 - if two triangles are similar, then the ratio of the altitudes is equal to the ratios of the corresponding sides

Theorem 7.9 - if two triangles are similar, then the ratio of the angle bisectors is equal to the ratios of the corresponding sides

Theorem 7.10 - if two triangles are similar, then the ratio of the medians is equal to the ratios of the corresponding sides

Theorem 7.11 (Triangle Angle Bisector Theorem) – an angle bisector of a triangle divides the opposite side into two segments whose lengths are proportional to their adjacent side

8.1: Geometric Mean

Geometric Mean – a number that can be placed in the denominator of one ratio and the numerator of another making the proportion equal

Theorem 8.1 – the altitude to the hypotenuse of a right triangle forms two triangles that are similar to each other and the original triangle

Theorem 8.2 – the length of the altitude to the hypotenuse of a right triangle is the geometric mean between the pieces of the hypotenuse formed by the altitude

Theorem 8.3 – the length of a leg of a right triangle is the geometric mean between the segment of the hypotenuse adjacent to it and the whole hypotenuse

8.2: The Pythagorean Theorem and Its Converse

Theorem 8.4 (Pythagorean Theorem) – A2 + B2 = C2

Theorem 8.5 (Converse of the Pythagorean Theorem) – if A2 + B2 = C2, then it is a right triangle

Pythagorean Triple – all three sides of a right triangle are whole numbers Pythagorean Inequalities Theorem – if A2 + B2 < C2, then it is an obtuse triangle

and if A2 + B2 > C2, then it is an acute triangle

8.3: Special Right Triangles

45-45-90 Triangle – has these as the measures of the interior angles Theorem 8.6 (45-45-90 Triangle Theorem) – the hypotenuse is equal to a leg

times the square root of 2 30-60-90 Triangle – has these as the measures of the interior angles Theorem 8.7 (30-60-90 Triangle Theorem) – the hypotenuse is equal to the short

leg times 2, and the long leg is equal to the short leg times the square root of 3

8.4: Trigonometry

Trigonometry - the study of triangle measurements Trigonometric Ratio – ratio of the lengths of two sides of a right triangle Opposite Leg – leg opposite the acute angle of a right triangle Adjacent Leg – leg adjacent to the acute angle of a right triangle Sine – the ratio of the opposite leg to the hypotenuse (opposite/hypotenuse) Cosine – the ratio of the adjacent leg to the hypotenuse (adjacent/hypotenuse) Tangent – the ratio of the opposite leg to the adjacent leg (opposite/adjacent)

Inverse Function – a method used to solve for the acute angles in a right triangle Inverse Tangent – if tan A = B, then tan-1 B = measure of angle A Inverse Sine – if sine A = B, then sine-1 B = measure of angle A Inverse Cosine – if cos A = B, then cos-1 B = measure of angle A

8.5: Angles of Elevation and Depression

Angle of Elevation – angle formed by a horizontal line and a line of sight to a point above the line

Angle of Depression – angle formed by a horizontal line and a line of sight to a point below the line

8.6: Law of Sines Theorem 8.8 (Law of Sines) – for any triangle with side lengths a, b, and c the

SinA/a = SinB/b = SinC/c Solving a Right Triangle – involves finding the measures of all the acute angles

and sides of a right triangle

8.7: Law of Cosines

Theorem 8.9: (Law of Cosines) – for any triangle with side lengths a, b, and c a2 = b2 + c2 – 2bccosA; b2 = a2 + c2 – 2accosB; c2 = a2 + b2 – 2abcosC

9.1: Reflections

Transformation – a change in the position, size, or shape of a figure Pre-Image – original figure, drawn in blue in the book Image – new figure, drawn in red in the book and identified with primes ( ` ) Isometry – a transformation that does not change the size or shape of the figure;

reflections, rotations, and translations are all isometries Reflection – transformation that moves a figure by flipping each point the same

distance over a line, much like an image in a mirror Line of Reflection - the line the figure is flipped over Reflection Across the X-Axis – will result in the y-coordinate flipping signs Reflection Across the Y-Axis – will result in the x-coordinate flipping signs Reflection Across the Line Y=X – will result in the x and y coordinates flipping

positions with each other Reflection Across the Line Y=-X – will result in the x and y coordinates flipping

positions with each other along with their signs Symmetry – the image coincides with the pre-image Line Symmetry – reflection across a line so that the image coincides with the pre-

image Line of Symmetry – splits a figure into two congruent halves so that it can be

folded onto itself Point of Symmetry - a point in common between both the pre-image and image

9.2: Translations

Translation – also called a slide is a transformation that moves all the points in a figure the same distance and direction

Composition - a transformation made up of successive transformations Two Reflections Theorem – two reflections across two parallel lines is equivalent

to a translation and the length of the translation is twice the distance between the parallel lines

Translation Theorem – any translation is equivalent to a composition of two reflections

Horizontal Translation – will result in only the x-coordinate changing Vertical Translation – will result in only the y-coordinate changing General Translation – will result in the x and y coordinates changing

9.3: Rotations

Rotation – transformation that moves each point of a figure the same distance around a fixed point

Center of Rotation - the fixed point Angle of Rotation - measure of the angle formed between corresponding points

and the center of rotation Rotation of 90 Degrees Around the Origin – will result in the flipping of the x and

y coordinates and the careful noting of whether they are positive or negative depending on the quadrant

Rotation of 180 Degrees Around the Origin – will result in the flipping of the x and y coordinates signs

Rotational Symmetry – being able to map a figure back onto itself in a rotation of 180 degrees or less

Theorem 9.1 – two reflections across intersecting lines is equivalent to a rotation with the center of rotation being the point of intersection of the lines and the angle of rotation being twice the measure of the angle formed by the lines where the rotation is occurring

Corollary 9.1 – reflecting an image successively in two perpendicular lines will result in a 180 degree rotation

Rotation Theorem - any rotation is equivalent to a composition of two reflections

9.4: Tessellations

Tessellation – repeating pattern that completely covers a plane with no gaps or overlaps, the measures of the angles at each vertex must equal 360 degrees

Regular Tessellation – formed by congruent regular polygons Uniform - the same arrangement of shapes and angles at each vertex Semi-regular Tessellation – formed by two or more different regular polygons,

with the same number of each polygon occurring in the same order at each vertex

Translation Symmetry – translated along a vector so that the image coincides with the pre-image

Frieze Pattern – has translation symmetry along a line Glide Reflection Symmetry – coincides with its image after a glide reflection

9.5: Dilations

Dilation – a transformation that produces similar figures through a reduction or enlargement

Reduction – image is smaller than the pre-image Enlargement – image is larger than the pre-image Scale Factor – length of the image divided by the length of the pre-image (r) Identify a dilation by its center point and scale factor Similarity Transformation - transformations that produce similar figures Theorem 9.2 - the length of any part of the image can be found by multiplying the

corresponding part of the pre-image by the scale factor Theorem 9.3 (Dilations in the Coordinate Plane) – if the center is the origin then

the resulting coordinates for the image will be the coordinates of the pre-image multiplied by the scale factor

9.6: Vectors

Vector – a quantity that has both length/magnitude and direction Initial Point – point where the vector begins (endpoint) Terminal Point – point where the vector ends (arrow) Magnitude – the length of the vector and is shown in absolute value symbols Direction – the angle a vector makes with a horizontal line Standard Position - the initial point of the vector is at the origin Component Form – lists the horizontal and vertical change of a vector < > Coordinate Notation – lists the horizontal and vertical change of a vector as a sum

or difference from X and Y in parenthesis Equal Vectors – have the same magnitude and direction Parallel Vectors – have the same or opposite directions Resultant Vector – represents the sum of two vectors Head to Tail Method – place the initial point of the second vector at the terminal

point of the first and the resultant vector will be from the initial point of the first to the terminal point of the second

Parallelogram Method – use the same initial point for both vectors and create a parallelogram by adding a copy of each vector at the other vector’s terminal point, and the resultant vector is a diagonal of the parallelogram formed

Scalar - a positive constant Scalar Multiplication - multiplying a vector by a scalar, which will result in a

change in magnitude but not direction for the vector

10.1: Circles and Circumference

Circle – a set of points a given distance from a fixed point Center - the fixed point Interior of a Circle – set of all points inside the circle Exterior of a Circle – set of all points outside the circle Radius – distance from the center to the circle Chord – a segment whose endpoints are on the circle Diameter – a chord that goes through the center Circumference – distance around a circle Circumference of a Circle - 2πR or πD Secant – a line that intersects the circle in two places Tangent – a line that intersects the circle in one place Point of Tangency – the one point a tangent intersects the circle Concentric Circles – coplanar circles with a common center Congruent Circles – have congruent radii Tangent Circles – coplanar circles that intersect at one point Common Tangents – line or segment that is tangent to two coplanar circles, you

can have anywhere from 0 to 4 common tangents

10.2: Measuring Angles and Arcs

Arc – unbroken part of a circle formed by the endpoints of an angle on a circle Central Angle – angle whose vertex is the center of the circle Minor Arc – less than 180 degrees, and named with two points Major Arc – greater than 180 degrees, and named with three points Semicircle – equals 180 degrees, and named with three points Measure of a Minor Arc – equals the measure of its central angle Measure of a Major Arc – equals 360 minus the related central angle Adjacent Arcs – arcs of the same circle that intersect at exactly one point Congruent Arcs – have the same measure Theorem 10.1 - in same or congruent circles, two arcs are congruent if and only if

the measure of their central angles are congruent Postulate 10.1 (Arc Addition Postulate) – larger arc equals the sum of the 2

adjacent arcs Arc Length – the portion of the circle between the endpoints of the arc, and is

found by using the following proportion; Arc Length/Circumference = Measure of the Arc/360 Degrees

10.3: Arcs and Chords

Theorem 10.2 – in same or congruent circles, congruent central angles have congruent chords, congruent chords have congruent arcs, and congruent arcs have congruent central angles

Inscribed – drawn within another figure, where the vertices of the inner figure touch the outer figure

Circumscribed – drawn around another figure, where the outer figure touches the vertices of the inner figure

Theorem 10.3 – if a radius or diameter is perpendicular to a chord, the it bisects the chord and its arc

Theorem 10.4 - in same or congruent circles, congruent chords are equidistant from the center

10.4: Inscribed Angles

Inscribed Angle – angle whose vertex lies on the circle Intercepted Arc – arc cut by the angle Theorem 10.5 (Measure of an Inscribed Angle) – 1/2 measure of the arc it cuts Theorem 10.6 - in same or congruent circles, if two inscribed angles cut the same

or congruent arcs then the angles are congruent Theorem 10.7 – if a right triangle is inscribed in a circle, then its hypotenuse is a

diameter of the circle Theorem 10.8 – if a quadrilateral is inscribed in a circle, then its opposite angles

are supplementary

10.5: Tangents

Tangent – a line that intersects the circle in one place Point of Tangency – the one point a tangent intersects the circle Theorem 10.9 – if a line is tangent to a circle, then it is perpendicular to the radius

drawn to the point of tangency Theorem 10.10 – if a line is perpendicular to the radius at a point on the circle,

then the line is tangent to the circle Theorem 10.11 – if two segments from the same exterior point are tangent to a

circle, then they are congruent

10.6: Secants, Tangents, and Angle Measures

Secant – a line that intersects the circle in two places Theorem 10.12 (Interior Angles) – if two secants or chords intersect inside a

circle then the measure of each angle equals 1/2 the sum of the arc it cuts and its vertical angle’s arc

Theorem 10.13 (Tangent Chord Angles) – if a tangent and a chord/secant intersect on a circle at the point of tangency, then the measure of the angle formed is 1/2 the measure of the intercepted arc

Theorem 10.14 (Exterior Angles) – if a tangent and a secant, or two tangents, or two secants intersect in the exterior of the circle, then the measure of the angle equals 1/2 the difference of the intercepted arcs

10.7: Special Segments in a Circle

Theorem 10.15 (Chord-Chord Product Theorem) – if two chords intersect inside a circle, then the product of the pieces of the first chord equals the product of the pieces of the second chord

Secant Segment – is a segment of the secant with at least one endpoint on the circle

External Secant Segment – secant segment on the exterior of the circle Theorem 10.16 (Secant-Secant Product Theorem) – if two secants intersect

outside the circle, then the product of the secant segment and external secant segment of one secant is equal to the product of the secant segment and external secant segment of the other secant

Tangent Segment – segment of the tangent with one endpoint on the circle Theorem 10.17 (Secant-Tangent Product Theorem) – if a secant and a tangent

intersect outside the circle, then the product of the secant segment and external secant segment of the secant is equal to the tangent segment squared

10.8: Equations of Circles

Theorem 11.7.1 (Standard Equation of a Circle) – (X – H)2 + (Y – K)2 = R2, where R is the radius of the circle, H is the x-coordinate of the center of the circle, and K is the y-coordinate of the center of the circle

To determine if a point falls on, inside, or outside a circle, simply plug in the X and Y values of the point

(X – H)2 + (Y – K)2 = R2, then the point lies on the circle (X – H)2 + (Y – K)2 < R2, then the point lies inside the circle (X – H)2 + (Y – K)2 > R2, then the point lies outside the circle

11.1: Areas of Parallelograms

Base of a Parallelogram – either pair of parallel sides Height of a Parallelogram – perpendicular distance between bases Area of a Parallelogram – BH, where B is the base and H is the perpendicular

distance between bases Area of a Square – S2, where S is the length of a side Area of a Rectangle – BH, where B is the base and H is the height

11.2: Areas of Triangles, Trapezoids, and Rhombi

Area of a Triangle – 1/2(BH), where B is the base and H is the perpendicular height

Area of a Trapezoid – 1/2(B1 + B2)*H, where B1 and B2 are the lengths of the bases and H is the perpendicular distance between the bases

Area of a Rhombus – 1/2(D1 * D2), where D1 and D2 are the lengths of the diagonals

Area of a Kite – 1/2(D1 * D2), where D1 and D2 are the lengths of the diagonals Postulate 11.1 - Congruent figures have equal areas

11.3: Areas of Regular Polygons and Circles

Center of a Regular Polygon – equidistant from the vertices

Apothem – the perpendicular distance from the center to a side Central Angle of a Regular Polygon – has its vertex at the center of the polygon

and its sides intersect with consecutive vertices Area of a Regular Polygon – 1/2AP, where A is the apothem and P is the perimeter Area of a Circle – πR2, where R is the radius

11.4: Areas of Composite Figures

Postulate 11.2 (Area Addition Postulate) – the area of a region is equal to the sum of its non-overlapping parts

Composite Figure – made up of simple shapes, such as triangles, rectangles, trapezoids and circles, and to find the area you must find the area of each individual shape

11.5: Geometric Probability and Area of Sectors

Probability – number of times a given event can occur/all possible outcomes Geometric Probability – probability of an event is based on a ratio of geometric

measures such as length or area Probability Involving Length – length of piece/length of whole Probability Involving Area – area of piece/area of whole Sector of a Circle – the region of the circle bounded by two radii and their

intercepted arc Area of a Sector – area of sector/area of circle = measure of the sector’s arc/360 Segment of a Circle – the region of a circle bounded by an arc and its chord Area of a Segment – (area of sector – area of triangle)

12.1: Solid Geometry

Solids – 3 dimensional figures Polyhedron – solid formed using polygons To name a polyhedron use the name of the shape of its base Base – two congruent polygons in parallel planes Faces – sides of the polyhedron Edges – segments formed by the intersection of two polygons Vertices – point where 3 or more faces meet Prism – type of polyhedron formed by two parallel congruent polygon bases with

the faces being parallelograms Pyramid – type of polyhedron formed by a polygon base with the faces being

triangles that meet at a common vertex Cylinder – solid formed by two parallel congruent circles as bases with a curved

surface connecting the bases Cone – solid formed using one circle as a base with a curved surface connecting

the base to a vertex Sphere – a solid made up of a set of points all a given distance from the center Cross Section – intersection of a 3-dimensional figure with a plane

Regular Polyhedron – all faces are congruent regular polygons Platonic Solids – the 5 regular polyhedron are tetrahedron (4 faces), cube (6

faces), octahedron (8 faces), dodecahedron (12 faces), icosahedron (20 faces)

12.2: Representation of 3-Dimensional Figures

Orthographic Drawing – shows six different views of a 3-dimensional figure Isometric Drawing – a 3-dimensional drawing showing 3 sides from a corner Perspective Drawing – non-vertical parallel lines are drawn to meet at a point Vanishing Point – the point a perspective drawing’s lines meet at Horizon – lines vanishing points appear on

12.3: Formulas in 3-Dimensions

Theorem 10.3.1 (Euler’s Theorem) – # faces + # vertices = # edges + 2 Diagonals of a Right Rectangular Prism - √L2 + W2 + H2

Space – the set of all points in 3-dimensions Distance Formula in 3-Dimensions - √(X2 – X1)2 + (Y2 – Y1)2 + (Z2 – Z1)2

Midpoint Formula in 3-Dimensions – (X1 + X2)/2, (Y1 + Y2)/2, (Z1 + Z2)/2

12.4: Surface Area of Prisms and Cylinders

Surface Area – sum of the area of the faces of a solid Lateral Faces – the faces of a solid that are not the bases Lateral Edge – edges formed by the intersection of two faces Lateral Area – the area of the lateral faces Right Prism – all the lateral faces are rectangles Oblique Prism – has at least one non-rectangular lateral face In this section we only find the surface area of right prisms and cylinders Lateral Area of a Right Prism – PH, where P is the perimeter of the base, and H

is the height between the bases Surface Area of a Right Prism – 2B + PH, where B is the area of the base, P is

the perimeter around the base, and H is the height between the bases Lateral Surface – curved surface connecting the two bases Axis of the Cylinder – segment with endpoints at the centers of the bases Right Cylinder – the axis is perpendicular to the bases Oblique Cylinder – the axis is not perpendicular to the bases Lateral Area of a Right Cylinder – CH, where C is the circumference of the base,

and H is the height between the bases Surface Area of a Right Cylinder – 2B + CH, where B is the area of the base, C is

the circumference of the base, and H is the height between the bases

12.5: Surface Area of Pyramids and Cones

Vertex of a Pyramid – the point opposite the base

Regular Pyramid – the base is a regular polygon, and the lateral faces are congruent isosceles triangles

Slant Height of a Pyramid – distance from the vertex to the midpoint of an edge of the base

Altitude of a Pyramid – perpendicular segment from the vertex to the base In this section we only find the surface area of right pyramids and cones Height of a Pyramid or Cone – the perpendicular distance from the base and the

vertex opposite it Lateral Area of a Right Pyramid - 1/2Pℓ, where P is the perimeter of the base, and

ℓ is the slant height of a lateral face Surface Area of a Right Pyramid – B + 1/2Pℓ, where B is the area of the base, P is

the perimeter of the base, and ℓ is the slant height of a lateral face Vertex of a Cone – the point opposite the base Axis of a Cone – segment with endpoints at the vertex and the center of the base Right Cone – the axis is perpendicular to the base Oblique Cone – the axis is not perpendicular to the base Slant Height of a Cone– distance from the vertex to a point on the edge of the

base Altitude of a Cone – perpendicular segment from the vertex to the base Lateral Area of a Right Cone – πRℓ, where R is the radius of the base, and ℓ is the

slant height of the lateral face Surface Area of a Right Cone – B + πRℓ, where B is the area of the base, R is the

radius of the base, and ℓ is the slant height of the lateral face

13.1: Volume of Prisms and Cylinders

Volume – the amount of space inside a solid measured in cubic units Cavalieri’s Principle – if two 3-dimensional figures have the same height and

cross sectional area at every level, then they have the same volume Volume of a Prism – BH, where B is the area of the base, and H is the

perpendicular height between the bases Volume of a Cylinder – BH, where B is the area of the base, and H is the

perpendicular height between the bases

13.2: Volume of Pyramids and Cones

Volume of a Pyramid – 1/3BH, where B is the area of the base, and H is the perpendicular height from the base to the opposite vertex

Volume of a Cone – 1/3πR2H, where R is the radius of the base, and H is the perpendicular height from the base to the opposite vertex

13.3: Spheres

Sphere – a solid made up of a set of points all a given distance from the center Radius of a Sphere – distance from the center to any point on the sphere

Hemisphere – half a sphere Great Circle – divides a sphere into two hemispheres Surface Area of a Sphere - 4πR2, where R is the radius Volume of a Sphere – 4/3πR3, where R is the radius