Math Handbook of Formulas, Processes and Tricks

Math Handbook of Formulas, Processes and Tricks

(mathguy.us)

Geometry

Prepared by: Earl L. Whitney, FSA, MAAA Version 4.2

August 26, 2023

Copyright 2010-2023, Earl Whitney, Reno NV. All Rights Reserved

Geometry Handbook Table of Contents

Page Description

Chapter 1: Basics

6

Points, Lines & Planes

7

Segments, Rays & Lines

8

Distance Between Points in 1 Dimension

8

Distances Between Collinear Points

9

Distance Between Points in 2 Dimensions

11

Partial Distances and Distance Equations

12

Distance Formula in "n" Dimensions

13

Angles

14

Types of Angles

Chapter 2: Proofs

16

Conditional Statements (Original, Converse, Inverse, Contrapositive)

17

Basic Properties of Algebra (Equality and Congruence, Addition and Multiplication)

18

Inductive vs. Deductive Reasoning

19

An Approach to Proofs

Chapter 3: Parallel and Perpendicular Lines

22

Parallel Lines and Transversals

23

Multiple Sets of Parallel Lines

24

Proving Lines are Parallel

25

Parallel and Perpendicular Lines in the Coordinate Plane

27

Proportional Segments

Chapter 4: Triangles - Basic

29

What Makes a Triangle?

31

Inequalities in Triangles

35

Types of Triangles (Scalene, Isosceles, Equilateral, Right)

37

Congruent Triangles (SAS, SSS, ASA, AAS, HL, CPCTC)

40

Centers of Triangles

42

Length of Height, Median and Angle Bisector

Chapter 5: Polygons

43

Polygons ? Basic (Definitions, Names of Common Polygons)

44

Polygons ? More Definitions (Definitions, Diagonals of a Polygon)

45

Interior and Exterior Angles of a Polygon

Cover art by Rebecca Williams, Twitter handle: @jolteonkitty

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Geometry Handbook Table of Contents

Page Description

Chapter 6: Quadrilaterals

46

Definitions of Quadrilaterals

47

Figures of Quadrilaterals

48

Amazing Property of Quadrilaterals

52

Characteristics of Parallelograms

53

Parallelogram Proofs (Sufficient Conditions)

54

Kites and Trapezoids

Chapter 7: Transformations

55

Introduction to Transformation

57

Reflection

59

Rotation

61

Translation

63

Compositions

65

Rotation About a Point Other than the Origin

Chapter 8: Similarity

68

Ratios Involving Units

69

Similar Polygons

70

Scale Factor of Similar Polygons

71

Dilations of Polygons

73

More on Dilation

74

Similar Triangles (SSS, SAS, AA)

75

Proportion Tables for Similar Triangles

78

Three Similar Triangles

Chapter 9: Right Triangles

80

Pythagorean Theorem

81

Pythagorean Triples

83

Special Triangles (45-45-90 Triangle, 30-60-90 Triangle)

85

Trigonometric Functions and Special Angles

86

Trigonometric Function Values in Quadrants II, III, and IV

87

Graphs of Trigonometric Functions

90

Vectors

91

Operating with Vectors

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Geometry Handbook Table of Contents

Page Description

Chapter 10: Circles

92

Parts of a Circle

93

Angles, Arcs, and Segments

94

Circle Vocabulary

95

Facts about Circles

95

Facts about Chords

97

Facts about Tangents

Chapter 11: Perimeter and Area

98

Perimeter and Area of a Triangle

100

More on the Area of a Triangle

101

Perimeter and Area of Quadrilaterals

102

Perimeter and Area of Regular Polygons

106

Circle Lengths and Areas

108

Area of Composite Figures

Chapter 12: Surface Area and Volume

111

Polyhedra

112

A Hole in Euler's Theorem

113

Platonic Solids

114

Prisms

116

Cylinders

118

Surface Area by Decomposition

119

Pyramids

121

Cones

123

Spheres

125

Similar Solids

127 Appendix A: Geometry Formulas 129 Appendix B: Trigonometry Formulas

131 Index

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Geometry Handbook Table of Contents

Useful Websites

Wolfram Math World ? Perhaps the premier site for mathematics on the Web. This site contains definitions, explanations and examples for elementary and advanced math topics. mathworld.

Mathguy.us ? Developed specifically for math students from Middle School to College, based on the author's extensive experience in professional mathematics in a business setting and in math tutoring. Contains free downloadable handbooks, PC Apps, sample tests, and more. mathguy.us

Broken Arrow, Oklahoma Standard Geometry Test ? A standardized Geometry test released by the state of Oklahoma. A good way to test your knowledge. pages/uploaded_files/Geometry%20Practice%20Test.pdf

Schaum's Outlines

An important student resource for any high school math student is a Schaum's Outline. Each book in this series provides explanations of the various topics in the course and a substantial number of problems for the student to try. Many of the problems are worked out in the book, so the student can see examples of how they should be solved.

Schaum's Outlines are available at , Barnes & Noble and other booksellers.

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Chapter 1

Geometry Points, Lines & Planes

Basic Geometry

Item

Illustration

Notation

Definition

Point Segment

A location in space. A straight path that has two endpoints.

Ray Line Plane

l or

m or

(points , , not linear)

A straight path that has one endpoint and extends infinitely in one direction.

A straight path that extends infinitely in both directions.

A flat surface that extends infinitely in two dimensions.

Collinear points are points that lie on the same line. Coplanar points are points that lie on the same plane.

In the figure at right: , , , , and are points.

l is a line m and n are planes.

In addition, note that: , , and are collinear points. , and are coplanar points. , and are coplanar points. Ray goes off in a southeast direction. Ray goes off in a northwest direction.

Together, rays and make up line l. Line l intersects both planes m and n.

Note: In geometric figures such as the one above, it is important to remember that, even though planes are drawn with edges, they extend infinitely in the 2 dimensions shown.

An intersection of geometric shapes is the set of points they share in common.

l and m intersect at point E. l and n intersect at point D. m and n intersect in line .

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Chapter 1

Geometry Segments, Rays & Lines

Basic Geometry

Some Thoughts About ...

Line Segments

Line segments are generally named by their endpoints, so the segment at right could be named either or .

Segment contains the two endpoints (A and B) and all points on line that are between them.

Congruent segments are segments of equal length.

A bisector splits a segment into two congruent (equal length) segments.

Rays

Rays are generally named by their single endpoint, called an initial point, and another point on the ray.

Ray contains its initial point A and all points on line in the direction of the arrow.

Rays and are not the same ray. If point O is on line and is between points A and B,

then rays and are called opposite rays. They have only point O in common, and together they make up line .

Lines

Lines are generally named by either a single script letter

(e.g., l) or by two points on the line (e.g.,. ).

A line extends infinitely in the directions shown by its arrows.

Lines are parallel if they are in the same plane and they

never intersect. Lines f and g, at right, are parallel.

Lines are perpendicular if they intersect at a 90 angle. A pair of perpendicular lines is always in the same plane. Lines

f and e, at right, are perpendicular. Lines g and e are also

perpendicular.

Lines are skew if they are not in the same plane and they

never intersect. Lines k and l, at right, are skew.

(Remember this figure is 3-dimensional.)

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Chapter 1

Geometry Distance Between Points

Basic Geometry

Distance measures how far apart two things are. The distance between two points can be measured in any number of dimensions, and is defined as the length of the line connecting the two points. Distance is always a positive number.

1-Dimension (line segment)

Distance - In one dimension, the distance between two points is determined simply by subtracting the coordinates of the points. If the endpoints are labeled, say A and B, then the length of segment AB is shown as AB.

Example 1.1: In this segment, the length of AB, i.e., AB, is calculated as: 5

A

B

2 .

Midpoint ? the point equidistant from each end of a line segment. That is, the midpoint is

halfway from one end of the segment to the other. To obtain the value of the midpoint, add the two end values and divide the result by 2.

Example 1.2: The midpoint of segment AB in Example 1.1 is:

.

Distances Between Collinear Points

Recall that collinear points are points on the same line.

A common problem in geometry is to split a line segment into parts based on some knowledge about the one or more of the parts.

Example 1.3: Find two possible lengths for CD if C, D, and E are collinear, and CE 15.8 cm and DE 3.5 cm.

It is helpful to use a line diagram when dealing with midpoint problems. There are two possible line diagrams for this problem: 1) D is between C and E, 2) E is between C and D.

In these diagrams, we show distances instead of point values:

Case 1

Case 2

15.8 3.5 .

15.8 3.5 .

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