Geometry Handbook PDF

Math Handbook of Formulas, Processes and Tricks (www.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 ...

Math Handbook of Formulas, Processes and Tricks (www.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 Version 4.2 Page 2 of 137 August 26, 2023  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 (45⁰-45⁰-90⁰ Triangle, 30⁰-60⁰-90⁰ 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 Version 4.2 Page 3 of 137 August 26, 2023  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 Version 4.2 Page 4 of 137 August 26, 2023  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.wolfram.com/ 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. www.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. www.baschools.org/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 Amazon.com, Barnes & Noble and other booksellers. Version 4.2 Page 5 of 137 August 26, 2023 Chapter 1 Basic Geometry Geometry Points, Lines & Planes Item Illustration Notation Definition Point A location in space. Segment A straight path that has two endpoints. Ray A straight path that has one endpoint and extends infinitely in one direction. A straight path that extends infinitely in Line l or both directions. m or A flat surface that extends infinitely in Plane (points , , two dimensions. not linear) 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. An intersection of geometric  Ray goes off in a northwest direction. shapes is the set of points they  Together, rays and make up line l. share in common.  Line l intersects both planes m and n. l and m intersect at point E. Note: In geometric figures such as the one above, it is l and n intersect at point D. important to remember that, even though planes are m and n intersect in line. drawn with edges, they extend infinitely in the 2 dimensions shown. Version 4.2 Page 6 of 137 August 26, 2023 Chapter 1 Basic Geometry Geometry Segments, Rays & Lines Some Thoughts About … Line Segments  Line segments are generally named by their endpoints, so the segment at right could be named either 𝐴𝐵 or 𝐵𝐴.  Segment 𝐴𝐵 contains the two endpoints (A and B) and all points on line ⃖𝐴𝐵⃗ 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 𝐴𝐵⃗ contains its initial point A and all points on line ⃖ ⃗ in the direction of the arrow. 𝐴𝐵  Rays 𝐴𝐵⃗ and 𝐵𝐴⃗ are not the same ray.  If point O is on line ⃖𝐴𝐵⃗ and is between points A and B, then rays 𝑂𝐴⃗ and 𝑂𝐵⃗ are called opposite rays. They ⃖ ⃗. have only point O in common, and together they make up line 𝐴𝐵 Lines  Lines are generally named by either a single script letter (e.g., l) or by two points on the line (e.g.,. 𝐴𝐵 ⃖ ⃗).  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 90⁰ 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.) Version 4.2 Page 7 of 137 August 26, 2023 Chapter 1 Basic Geometry Geometry Distance Between Points 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 2 𝟕. A B 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. 𝟑 Example 1.2: The midpoint of segment AB in Example 1.1 is:. 𝟐 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 𝑥 15.8 3.5 𝟏𝟐. 𝟑 𝐜𝐦 𝑥 15.8 3.5 𝟏𝟗. 𝟑 𝐜𝐦 Version 4.2 Page 8 of 137 August 26, 2023 Chapter 1 Basic Geometry 2-Dimensions Distance – In two dimensions, the distance between two points can be calculated by considering the line between them to be the hypotenuse of a right triangle. To determine the length of this line:  Calculate the difference in the 𝑥-coordinates of the points  Calculate the difference in the 𝑦-coordinates of the points  Use the Pythagorean Theorem. This process is illustrated below, using the variable “d” for distance. Example 1.4: Find the distance between (-1,1) and (2,5). Based on the illustration to the left: x‐coordinate difference: 2 1 3. y‐coordinate difference: 5 1 4. Then, the distance is calculated using the formula: d 3 4 9 16 25 We get d 25, so d √25 𝟓 If we define two points generally as (x1, y1) and (x2, y2), then the 2-dimensional distance formula would be: distance x x y y. Midpoint – To obtain the value of the midpoint in two or more dimensions, add the corresponding coordinates of the endpoints and divide each result by 2. If you are given the value of the midpoint and asked for the coordinates of an endpoint, you may choose to calculate a vector, which in this case is simply the difference between two points. Example 1.5: Find the distance between P 2, 3 and Q 3, 15. The formula for the distance between points is: d 𝑥 𝑥 𝑦 𝑦 Let point 1 be P 2, 3 , and let point 2 be Q 3, 15. Then, d 3 2 15 3 √5 12 √169 𝟏𝟑 Note that 5-12-13 is a Pythagorean Triple. Version 4.2 Page 9 of 137 August 26, 2023 Chapter 1 Basic Geometry Example 1.6: The midpoint of segment AD is 1, 2. Point A has coordinates 3, 3 and point D has coordinates 𝑥, 7. It is helpful to use a line diagram when dealing with midpoint problems. Label the endpoints and midpoint, and identify the coordinates of each: The difference between points 𝐀 and M can be expressed in two dimensions as a vector using “〈 〉” instead of “ ”. Let’s find the difference (note: “difference” implies subtraction). 1, 2 Point 𝐌 3, 3 Point 𝐀 〈 2, 5〉 Difference vector (difference between the two points) The difference vector can then be applied to the midpoint to get the coordinates of point 𝐃. If I can get from A to M by moving 〈 2, 5〉, then I can get from M to D by moving 〈 2, 5〉. 1, 2 Point 𝐌 〈 2, 5〉 Difference vector 𝟏, 7 Point D. Therefore, we conclude that 𝒙 𝟏. Note that the 𝑦-value of point 𝐃 in the solution, 7, matches the 𝑦-value of point 𝐃 in the statement of the problem. Example 1.7: Find the value of 𝑦 if AC 3𝑦 5, CB 4𝑦 1, AB 9𝑦 12, and C lies between A and B. The line diagram is crucial for this problem. It must be drawn with A and B as endpoints and C between them. Based on the diagram, we have: 3𝑦 5 4𝑦 1 9𝑦 12 7𝑦 4 9𝑦 12 16 2𝑦 𝟖 𝒚 Version 4.2 Page 10 of 137 August 26, 2023 Chapter 1 Basic Geometry Partial Distances and Distance Equations In order to find a distance part-way between two points, we need to interpolate between the beginning and end points. We must calculate the portion of the distance covered at the desired time, and then interpolate between the start and end points. Let 𝑘 be the factor, representing the portion of the total distance that is of interest to us. 𝑘 is usually given in terms of time, e.g., after 3 hours of a 10-hour journey. In general, elapsed time 𝑘. total time The formula for the interpolation, then, is: desired point 𝑘 ∙ ending point 1 𝑘 ∙ starting point This interpolation formula works for any number of dimensions, taking each coordinate separately. Example 1.8: A boat begins a journey at location 2, 5 on a grid and heads directly for point 10, 15 on the same grid. It is estimated that the trip will take 10 hours if the boat travels in a straight line. At what point of the grid is the boat after 3 hours? Start at: 2, 5 End at: 10, 15 3 hours → 𝑘 0.3 of the 10 hour period.  This is the factor for the endpoint: 10, 15.  The staring point, 2, 5 gets a factor of 1 0.3 0.7. The factors must always add to 1. Ordered pair @ 𝑡 3 hours is: 2, 5 ∙ 0.7 10, 15 ∙ 0.3 𝟒. 𝟒, 𝟖. 𝟎 Note: an alternative method would be to develop separate equations for the 𝑥-variable and 𝑦-variable in terms of time, the 𝑡-variable. These are called parametric equations, and 𝑡 is the parameter in the equations. For this problem, the parametric equations would be: 𝑡 variable start end start ∙ period length in years 𝑡 𝑥 2 10 2 ∙ 2 0.8𝑡 10 𝑡 𝑦 5 15 5 ∙ 5 𝑡 10 Note that the 10 in the denominator of these equations is the length of time, in hours, separating the starting point and the ending point. Solve for the required ordered pair by substituting 𝑡 3 into these equations. Version 4.2 Page 11 of 137 August 26, 2023 Chapter 1 Basic Geometry Geometry ADVANCED Distance Formula in “n” Dimensions The distance between two points can be generalized to “n” dimensions by successive use of the Pythagorean Theorem in multiple dimensions. To move from two dimensions to three dimensions, we start with the two-dimensional formula and apply the Pythagorean Theorem to add the third dimension. 3 Dimensions Consider two 3-dimensional points (x1, y1, z1) and (x2, y2, z2). Consider first the situation where the two z-coordinates are the same. Then, the distance between the points is 2-dimensional, i.e., d 𝑥 𝑥 𝑦 𝑦. We then add a third dimension using the Pythagorean Theorem: distance d z z distance x x y y z z distance x x y y z z And, finally the 3-dimensional difference formula: distance x x y y z z n Dimensions Using the same methodology in “n” dimensions, we get the generalized n-dimensional difference formula (where there are n terms beneath the radical, one for each dimension): distance x x y y z z ⋯ w w Or, in higher level mathematical notation: The distance between two points A a ,a ,… ,a and 𝐵 b ,b ,… ,b is 𝑑 𝐴, 𝐵 |𝐴 𝐵| 𝑎 𝑏 Version 4.2 Page 12 of 137 August 26, 2023 Chapter 1 Basic Geometry Geometry Angles Parts of an Angle An angle consists of two rays with a common endpoint (or, initial point).  Each ray is a side of the angle.  The common endpoint is called the vertex of the angle. Naming Angles Angles can be named in one of two ways:  Point-vertex-point method. In this method, the angle is named from a point on one ray, the vertex, and a point on the other ray. This is the most unambiguous method of naming an angle, and is useful in diagrams with multiple angles sharing the same vertex. In the above figure, the angle shown could be named ∠BAC or ∠CAB.  Vertex method. In cases where it is not ambiguous, an angle can be named based solely on its vertex. In the above figure, the angle could be named ∠A. Measure of an Angle There are two conventions for measuring the size of an angle:  In degrees. The symbol for degrees is ⁰. There are 360⁰ in a full circle. The angle above measures approximately 360 8 45⁰ (one-eighth of a circle).  In radians. There are 2𝜋 radians in a complete circle. The angle above measures approximately radians. Some Terms Relating to Angles Angle interior is the area between the rays. Angle exterior is the area not between the rays. Adjacent angles are angles that share a ray for a side. ∠BAD and ∠DAC in the figure at right are adjacent angles. Congruent angles are angles with the same measure. Angle bisector is a ray that divides the angle into two congruent angles. Ray AD⃗ bisects ∠BAC in the figure at right. Version 4.2 Page 13 of 137 August 26, 2023 Chapter 1 Basic Geometry Geometry Types of Angles C A B D Supplementary Angles Complementary Angles Angles A and B are supplementary. Angles C and D are complementary. Angles A and B form a linear pair. 𝑚∠𝐶 𝑚∠𝐷 90⁰ 𝑚∠𝐴 𝑚∠𝐵 180⁰ Angles which are opposite each other when two lines cross are vertical angles. Angles E and G are vertical angles. F Angles F and H are vertical angles. E G H 𝑚∠𝐸 𝑚∠𝐺 𝑎𝑛𝑑 𝑚∠𝐹 𝑚∠𝐻 In addition, each angle is supplementary to the two angles adjacent to it. For example: Vertical Angles Angle E is supplementary to Angles F and H. An acute angle is one that is less than 90⁰. In the illustration above, angles E and G are acute angles. A right angle is one that is exactly 90⁰. Acute Obtuse An obtuse angle is one that is greater than 90⁰. In the illustration above, angles F and H are obtuse angles. A straight angle is one that is exactly 180⁰. Right Straight Version 4.2 Page 14 of 137 August 26, 2023 Chapter 1 Basic Geometry Example 1.9: Two angles are complementary. The measure of one angle is 21° more than twice the measure of the other angle. Find the measures of the angles. Drawing the situation described in the problem is often helpful. Let the two angles be called angle A and angle B. Let’s rewrite the problem in terms of these two angles. Angles A and B are complementary. 𝑚∠A 21° 2 𝑚∠B. Let the measures of the angles be represented by the names of the angles. Then, A B 90° 2A 2B 180° A B 90° A 21° 2B A 2B 21° 67° B 90° 3A 201° 𝐁 𝟐𝟑° 𝐀 𝟔𝟕° The measures of the two angles then, are, 𝟔𝟕° and 𝟐𝟑° Example 1.10: If m∠BGC 16x 4° and m∠CGD 2x 13°, find the value of 𝑥 so that ∠BGD is a right angle. ∠BGD is a right angle (i.e., m∠BGD 90°. Then, 16𝑥 4° 2𝑥 13° 90° 18𝑥 9° 90° 18𝑥 81° 𝒙 𝟒. 𝟓° Example 1.11: Find 𝑚∠1 if ∠1 is complementary to ∠2, ∠2 is supplementary to ∠3, and 𝑚∠3 126°. Let’s turn this into equations because the English is confusing. 𝑚∠1 𝑚∠2 90° (complementary) 𝑚∠2 𝑚∠3 180° (supplementary) 𝑚∠3 126° Working with these equations from bottom to top, we get: 𝑚∠3 126° 𝑚∠2 𝑚∠3 𝑚∠2 126° 180°, so 𝑚∠2 54° 𝑚∠1 𝑚∠2 𝑚∠1 54° 90° so 𝒎∠𝟏 𝟑𝟔° Version 4.2 Page 15 of 137 August 26, 2023 Chapter 2 Proofs Geometry Conditional Statements A conditional statement contains both a hypothesis and a conclusion in the following form: If hypothesis, then conclusion. Statements linked For any conditional statement, it is possible to create three related below by red arrows conditional statements, as shown below. In the table, p is the hypothesis must be either both of the original statement and q is the conclusion of the original statement. true or both false. Example Type of Conditional Statement Statement is: Original Statement: If p, then q. (𝒑 → 𝒒)  Example: If a number is divisible by 6, then it is divisible by 3. TRUE  The original statement may be either true or false. Converse Statement: If q, then p. (𝒒 → 𝒑)  Example: If a number is divisible by 3, then it is divisible by 6. FALSE  The converse statement may be either true or false, and this does not depend on whether the original statement is true or false. Inverse Statement: If not p, then not q. (~𝒑 → ~𝒒)  Example: If a number is not divisible by 6, then it is not divisible by 3. FALSE  The inverse statement is always true when the converse is true and false when the converse is false. Contrapositive Statement: If not q, then not p. (~𝒒 → ~𝒑)  Example: If a number is not divisible by 3, then it is not divisible by 6. TRUE  The Contrapositive statement is always true when the original statement is true and false when the original statement is false. Note also that:  When two statements must be either both true or both false, they are called equivalent statements. o The original statement and the contrapositive are equivalent statements. o The converse and the inverse are equivalent statements.  If both the original statement and the converse are true, the phrase “if and only if” (abbreviated “iff”) may be used. For example, “A number is divisible by 3 iff the sum of its digits is divisible by 3.” Version 4.2 Page 16 of 137 August 26, 2023 Chapter 2 Proofs Geometry Basic Properties of Algebra Properties of Equality and Congruence. Definition for Equality Definition for Congruence Property For any geometric elements a, b and c. For any real numbers a, b, and c: (e.g., segment, angle, triangle) Reflexive Property 𝑎 𝑎 𝑎≅𝑎 Symmetric Property 𝐼𝑓 𝑎 𝑏, 𝑡ℎ𝑒𝑛 𝑏 𝑎 𝐼𝑓 𝑎 ≅ 𝑏, 𝑡ℎ𝑒𝑛 𝑏 ≅ 𝑎 Transitive Property 𝐼𝑓 𝑎 𝑏 𝑎𝑛𝑑 𝑏 𝑐, 𝑡ℎ𝑒𝑛 𝑎 𝑐 𝐼𝑓 𝑎 ≅ 𝑏 𝑎𝑛𝑑 𝑏 ≅ 𝑐, 𝑡ℎ𝑒𝑛 𝑎 ≅ 𝑐 If 𝑎 𝑏, then either can be If 𝑎 ≅ 𝑏, then either can be Substitution Property substituted for the other in any substituted for the other in any equation (or inequality). congruence expression. More Properties of Equality. For any real numbers a, b, and c: Property Definition for Equality 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 𝑎 ∙ 𝑐 𝑏∙𝑐 Division Property 𝐼𝑓 𝑎 𝑏 𝑎𝑛𝑑 𝑐 0, 𝑡ℎ𝑒𝑛 𝑎 𝑐 𝑏 𝑐 Properties of Addition and Multiplication. For any real numbers a, b, and c: Property Definition for Addition Definition for Multiplication Commutative Property 𝑎 𝑏 𝑏 𝑎 𝑎∙𝑏 𝑏∙𝑎 Associative Property 𝑎 𝑏 𝑐 𝑎 𝑏 𝑐 𝑎∙𝑏 ∙𝑐 𝑎∙ 𝑏∙𝑐 Distributive Property 𝑎∙ 𝑏 𝑐 𝑎∙𝑏 𝑎∙𝑐 Version 4.2 Page 17 of 137 August 26, 2023 Chapter 2 Proofs Geometry Inductive vs. Deductive Reasoning Inductive Reasoning Inductive reasoning uses observation to form a hypothesis or conjecture. The hypothesis can then be tested to see if it is true. The test must be performed in order to confirm the hypothesis. Example: Observe that the sum of the numbers 1 to 4 is 4 ∙ 5/2 and that the sum of the numbers 1 to 5 is 5 ∙ 6/2. Hypothesis: the sum of the first n numbers is 𝑛 ∗ 𝑛 1 /2. Testing this hypothesis confirms that it is true. Deductive Reasoning Deductive reasoning argues that if something is true about a broad category of things, it is true of an item in the category. Example: All birds have beaks. A pigeon is a bird; therefore, it has a beak. There are two key types of deductive reasoning of which the student should be aware:  Law of Detachment. Given that 𝒑 → 𝒒, if p is true then q is true. In words, if one thing implies another, then whenever the first thing is true, the second must also be true. Example 2.1: Start with the statement: “If a living creature is human, then it has a brain.” Then because you are human, we can conclude that you have a brain.  Syllogism. Given that 𝒑 → 𝒒 and 𝒒 → 𝒓, we can conclude that 𝒑 → 𝒓. This is a kind of transitive property of logic. In words, if one thing implies a second and that second thing implies a third, then the first thing implies the third. Example 2.2: Start with the statements: “If my pencil breaks, I will not be able to write,” and “if I am not able to write, I will not pass my test.” Then I can conclude that “If my pencil breaks, I will not pass my test.” Version 4.2 Page 18 of 137 August 26, 2023 Chapter 2 Proofs Geometry An Approach to Proofs Learning to develop a successful proof is one of the key skills students develop in geometry. The process is different from anything students have encountered in previous math classes, and may seem difficult at first. Diligence and practice in solving proofs will help students develop reasoning skills that will serve them well for the rest of their lives. Requirements in Performing Proofs  Each proof starts with a set of “givens,” statements that you are supplied and from which you must derive a “conclusion.” Your mission is to start with the givens and to proceed logically to the conclusion, providing reasons for each step along the way.  Each step in a proof builds on what has been developed before. Initially, you look at what you can conclude from the” givens.” Then as you proceed through the steps in the proof, you are able to use additional things you have concluded based on earlier steps.  Each step in a proof must have a valid reason associated with it. So, each statement in the proof must be furnished with an answer to the question: “Why is this step valid?” Tips for Successful Proof Development  At each step, think about what you know and what you can conclude from that information. Do this initially without regard to what you are being asked to prove. Then look at each thing you can conclude and see which ones move you closer to what you are trying to prove.  Go as far as you can into the proof from the beginning. If you get stuck, …  Work backwards from the end of the proof. Ask yourself what the last step in the proof is likely to be. For example, if you are asked to prove that two triangles are congruent, try to see which of the several theorems about this is most likely to be useful based on what you were given and what you have been able to prove so far.  Continue working backwards until you see steps that can be added to the front end of the proof. You may find yourself alternating between the front end and the back end until you finally bridge the gap between the two sections of the proof.  Don’t skip any steps. Some things appear obvious, but actually have a mathematical reason for being true. For example, 𝑎 𝑎 might seem obvious, but “obvious” is not a valid reason in a geometry proof. The reason for 𝑎 𝑎 is a property of algebra called the “reflexive property of equality.” Use mathematical reasons for all your steps. Version 4.2 Page 19 of 137 August 26, 2023 Chapter 2 Proofs Proof examples (may use information presented later in this handbook) Example 2.3: Given: 𝑚∠1 𝑚∠3 180°. Prove: ∠2 ≅ ∠3. Recall that congruent angles have the same measure. Step Statement Reason 1 𝑚∠1 𝑚∠3 180° Given. ∠1 and ∠3 are supplementary. If the sum of two angles is 180°, then the 2 angles are supplementary. 3 ∠1 and ∠2 form a linear pair. Diagram. 4 ∠1 and ∠2 are supplementary. If two angles form a linear pair, then the angles are supplementary. 5 If two angles are supplementary to the same ∠2 ≅ ∠3 angle, then they are congruent. Example 2.4: Given: KJ ≅ MK, J is the midpoint of HK. Prove: HJ ≅ MK. Recall that congruent segments have the same measure. Thought process. Based on the givens, it appears that the three segments identified in the diagram are all congruent. That is, 𝐻𝐽 ≅ 𝐾𝐽 ≅ 𝑀𝐾. We need to work from the congruence we are given to the one we want to prove by considering how the segments relate to each other one pair at a time. Step Statement Reason 𝐾𝐽 ≅ 𝑀𝐾 1 Given 𝐽 is the midpoint of 𝐻𝐾 2 𝐾𝐽 ≅ 𝐻𝐽 A midpoint creates two congruent segments. Transitive property of congruence (in this case, 3 𝐻𝐽 ≅ 𝑀𝐾 two segments that are each congruent to a third segment are congruent to each other). Note: purple text in the proof is explanatory and is not required to complete the proof. Version 4.2 Page 20 of 137 August 26, 2023 Chapter 2 Proofs Example 2.5: Given: ∠𝐻 ≇ ∠𝐾. Prove: ∆𝐽𝐻𝐾 is not isosceles with base 𝐻𝐾. Note: the " ≇ " symbol means “is not congruent to”. We will use proof by contradiction on this problem. In proof by contradiction, we assume that the opposite of the conclusion is true, then show that is impossible. This implies that the original assumption is false, so its opposite (what we want to prove) must be true. Step Statement Reason 1 ∠𝐻, ∠𝐾 not congruent Given Assume ∆𝐽𝐻𝐾 is isosceles with base Assumption intended to lead to a 2 𝐻𝐾. contradiction. 3 𝐽𝐾 𝐽𝐻 Euclid’s definition of isosceles triangle. 4 𝐽𝐾 ≅ 𝐽𝐻 Definition of congruent segments. Angles opposite congruent sides in a 5 ∠𝐻 ≅ ∠𝐾 triangle are congruent. 6 Contradiction We are given ∠𝐻, ∠𝐾 are not congruent. 7 ∆𝐽𝐻𝐾 is not isosceles with base 𝐻𝐾. Assumption in Step 2 must be false. Additional proofs are provided throughout this handbook. Version 4.2 Page 21 of 137 August 26, 2023 Chapter 3 Parallel and Perpendicular Lines Geometry Parallel Lines and Transversals Transversal Alternate: refers to angles that are on opposite sides of the transversal. A B Consecutive: refers to angles that are C D on the same side of the transversal. Parallel Lines Interior: refers to angles that are F between the parallel lines. E H Exterior: refers to angles that are G outside the parallel lines. Corresponding Angles Corresponding Angles are angles in the same location relative to the parallel lines and the transversal. For example, the angles on top of the parallel lines and left of the transversal (i.e., top left) are corresponding angles. Angles A and E (top left) are Corresponding Angles. So are angle pairs B and F (top right), C and G (bottom left), and D and H (bottom right). Corresponding angles are congruent. Alternate Interior Angles Angles D and E are Alternate Interior Angles. Angles C and F are also alternate interior angles. Alternate interior angles are congruent. Alternate Exterior Angles Angles A and H are Alternate Exterior Angles. Angles B and G are also alternate exterior angles. Alternate exterior angles are congruent. Consecutive Interior Angles Angles C and E are Consecutive Interior Angles. Angles D and F are also consecutive interior angles. Consecutive interior angles are supplementary. Note that angles A, D, E, and H are congruent, and angles B, C, F, and G are congruent. In addition, each of the angles in the first group are supplementary to each of the angles in the second group. Version 4.2 Page 22 of 137 August 26, 2023 Chapter 3 Parallel and Perpendicular Lines Geometry Multiple Sets of Parallel Lines Two Transversals Sometimes, the student is presented two sets of intersecting parallel lines, as shown above. Note that each pair of parallel lines is a set of transversals to the other set of parallel lines. A B I J C D K L E F M N G H O P In this case, the following groups of angles are congruent:  Group 1: Angles A, D, E, H, I, L, M and P are all congruent.  Group 2: Angles B, C, F, G, J, K, N, and O are all congruent.  Each angle in the Group 1 is supplementary to each angle in Group 2. Some Examples: In the diagram above (Two Transversals), with two pairs of parallel lines, what types of angles are identified and what is their relationship to each other? Example 3.1: ∠𝐷 and ∠𝐼. These angles are alternate interior angles; they are congruent. Example 3.2: ∠𝐶 and ∠𝐽. These angles are alternate exterior angles; they are congruent. Example 3.3: ∠𝐽 and ∠𝑁. These angles are corresponding angles; they are congruent. Example 3.4: ∠𝐹 and ∠𝑀. These angles are consecutive interior angles; they are supplementary. Example 3.5: ∠𝐺 and ∠𝐿. These angles do not have a name, but they are supplementary. Version 4.2 Page 23 of 137 August 26, 2023 Chapter 3 Parallel and Perpendicular Lines Geometry Proving Lines are Parallel The properties of parallel lines cut by a transversal can be used to prove two lines are parallel. Corresponding Angles If two lines cut by a transversal have congruent corresponding angles, then the lines are parallel. Note that there are 4 sets of corresponding angles. Alternate Interior Angles If two lines cut by a transversal have congruent alternate interior angles congruent, then the lines are parallel. Note that there are 2 sets of alternate interior angles. Alternate Exterior Angles If two lines cut by a transversal have congruent alternate exterior angles, then the lines are parallel. Note that there are 2 sets of alternate exterior angles. Consecutive Interior Angles If two lines cut by a transversal have supplementary consecutive interior angles, then the lines are parallel. Note that there are 2 sets of consecutive interior angles. Version 4.2 Page 24 of 137 August 26, 2023 Chapter 3 Parallel and Perpendicular Lines Geometry Parallel and Perpendicular Lines in the Coordinate Plane Parallel Lines Two lines are parallel if their slopes are equal.  In 𝑦 𝑚𝑥 𝑏 form, if the values of 𝑚 are the same. Example 3.6: 𝑦 2𝑥 3 and 𝑦 2𝑥 1  In Standard Form, if the coefficients of 𝑥 and 𝑦 are proportional between the equations. Example 3.7: 3𝑥 2𝑦 5 and 6𝑥 4𝑦 7  Also, if the lines are both vertical (i.e., their slopes are undefined). Example 3.8: 𝑥 3 and 𝑥 2 Perpendicular Lines Two lines are perpendicular if the product of their slopes is 𝟏. That is, if the slopes have different signs and are multiplicative inverses.  In 𝑦 𝑚𝑥 𝑏 form, the values of 𝑚 multiply to get 1.. Example 3.9: 𝑦 6𝑥 5 and 𝑦 𝑥 3  In Standard Form, if you add the product of the x-coefficients to the product of the y- coefficients and get zero. Example 3.10: 4𝑥 6𝑦 4 and 3𝑥 2𝑦 5 because 4∙3 6∙ 2 0  Also, if one line is vertical (i.e., 𝑚 is undefined) and one line is horizontal (i.e., 𝑚 0). Example 3.11: 𝑥 6 and 𝑦 3 Version 4.2 Page 25 of 137 August 26, 2023 Chapter 3 Parallel and Perpendicular Lines Example 3.12: Write the equation of the perpendicular bisector of CD if C 4, 3 and D 8, 9. Line containing 𝐶𝐷 : 9 3 12 𝑚 3 8 47 4 Midpoint of 4, 3 and 8, 9 is halfway between them: 6, 3 Perpendicular bisector: Slope is the “negative reciprocal” of the slope of ⃖𝐶𝐷⃗ because the lines are perpendicular. Also, 6, 3 is a point on the perpendicular bisector. 𝑚 𝟏 𝟏 𝟏 Equation: 𝒚 𝟑 𝒙 𝟔 or 𝒚 𝒙 𝟔 𝟑 or 𝒚 𝒙 𝟓 𝟑 𝟑 𝟑 point-slope form ℎ-𝑘 form slope-intercept form Example 3.13: Write an equation of the line that can be used to calculate the distance between 4, 3 and the line 𝑦 𝑥 9. The distance between a point and a line is the length of the segment connecting the point to the line at a right angle. See the diagram to the right. So, this question is asking for the equation of the line perpendicular to 𝑦 𝑥 9 that contains the point 4, 3 , but is not asking us to calculate the distance. The perpendicular line will have a slope that is the opposite reciprocal of the original line: 1 7 𝑚 2 2 7 Then, the equation of the perpendicular line (in ℎ-𝑘 form) is: 𝟕 𝒚 𝒙 𝟒 𝟑 𝟐 Note: If we were asked to calculate the distance between Point A and the line 𝑦 𝑥 9, we would first need to find Point B at the intersection of the two lines shown, and then measure the distance between the two points using the distance formula. Version 4.2 Page 26 of 137 August 26, 2023 Chapter 3 Parallel and Perpendicular Lines Geometry Proportional Segments Parallel Line in a Triangle A line is parallel to one side of a triangle iff it divides the other two sides proportionately. This if-and-only-if statement breaks down into the following two statements:  If a line (or ray or segment) is parallel to one side of a triangle, then it divides the other two sides proportionately.  If a line (or ray or segment) divides two sides of a triangle proportionately, then it is parallel to the third side. In the diagram to the right, we see that 𝐴𝐵 ∥ 𝐸𝐷. We can conclude that: and as well as a number of other equivalent proportion equalities. Conversely, if we knew one of the proportions above, but were not given that the segments were parallel, we could conclude that 𝐴𝐵 ∥ 𝐸𝐷 because of the equal proportions. Example 3.14: Determine whether 𝐴𝐵 ∥ 𝐸𝐷 in the diagram to the right. Let’s check the proportions. Is ? 𝐶𝐸 12 3 𝐸𝐴 6 3 𝐶𝐷 8 2 𝐷𝐵 4 2 Since the proportions of the two sides are equal, we can conclude that 𝑨𝑩 ∥ 𝑬𝑫. Three or More Parallel Lines Three or more parallel lines divide any transversals proportionately. In the diagram to the right, we see that the three horizontal lines (or rays or segments) are parallel. We can conclude that: and. The converse of this is not true. That is, if three or more lines divide transversals into proportionate parts, it is not necessarily true that the lines are parallel. Version 4.2 Page 27 of 137 August 26, 2023 Chapter 3 Parallel and Perpendicular Lines Example 3.15: Given that the three horizontal lines in the diagram to the right are parallel, what is the values of 𝑥? The three parallel horizontal lines in the diagram divide the vertical lines into proportional segments. 25 30 10 𝑥 25𝑥 300 𝒙 𝟏𝟐 Angle Bisector An angle bisector in a triangle divides the opposite sides into segments that are proportional to the adjacent sides. In the diagram to the right, we see that ∠𝐷 is bisected, crea ng segments 𝐴𝐵 and 𝐵𝐶 opposite ∠𝐷. We can conclude that: and. The converse of this is also true. That is, if a line (or ray or segment) through a vertex of a triangle splits the opposite side into segments that are proportional to the adjacent sides, then, that line (or ray or segment) bisects the vertex angle. That is, if the above proportions are true, then 𝐷𝐵 bisects ∠𝐷. Example 3.16: Find the value of 𝑥 in the diagram. An angle bisector in a triangle divides the opposite sides into segments that are proportional to the adjacent sides. So, 18 𝑥 3 𝑥 10 𝑥 𝑥 3 18 ∙ 10 𝑥 3𝑥 180 𝑥 3𝑥 180 0 𝑥 12 𝑥 15 0 → 𝑥 12, 15 If 𝑥 15, we have negative side lengths, so we discard the solution 𝑥 15. If 𝑥 12, the sides of ∆𝐵𝐴𝐷 would be 18, 15, 22 , which makes a valid triangle. Conclude: 𝒙 𝟏𝟐. Version 4.2 Page 28 of 137 August 26, 2023 Chapter 4 Triangles - Basic Geometry What Makes a Triangle? Definition – A triangle is a plane figure with three sides and three angles.  Draw three points that are not on the same line, connect them, and you have a triangle. The three points you started with are called vertices.  Three points determine a plane, so a triangle must have all of its parts on the same plane. Parts of a Triangle  Vertices – the points where the sides intersect. In the diagram to the right, the vertices are the red points. Vertices are typically labeled with capital letters.  Legs – the sides of a triangle are also called the triangle’s legs. In diagrams, the lengths of the legs are often represented by lower case letters corresponding to the angles opposite them.  Angles (interior angles) – the angles formed at each vertex are the triangle’s angles. In the diagram above, the triangle has interior angles ∠𝐴, ∠𝐵, ∠𝐶 indicated by the green arcs at the vertices. These angles could be named in various ways, for example: o ∠𝐴 ∠𝐵𝐴𝐶 ∠𝐶𝐴𝐵. o Naming the angle with a single vertex is acceptable if there is no ambiguity about which angle is being referenced, e.g., ∠𝐴. o If any ambiguity exists as to which angle is being referenced, the angle must be named using three points: two of the points must be on the sides enclosing the angle and the vertex must be in the middle, e.g., ∠𝐵𝐴𝐶 or ∠𝐶𝐴𝐵. o Alternatively, an angle may be named with a letter or symbol next to its arc.  Altitudes – line segments from each vertex to the opposite side of the triangle that are perpendicular to that opposite side. In the diagram below left, an altitude is labeled h.  Medians – line segments from each vertex to the midpoint of the opposite side of the triangle. In the diagram below right, a median is labeled m. Version 4.2 Page 29 of 137 August 26, 2023 Chapter 4 Triangles - Basic Sum of Interior Angles The sum of the interior angles of a triangle is 180°. If two of the interior angles in a triangle have known measures, the measure of the third can be easily calculated. For example, in the diagram to the right, if 𝑚∠𝐴 and 𝑚∠𝐵 are known, 𝑚∠𝐶 can be calculated as: 𝑚∠𝐶 180° 𝑚∠𝐴 𝑚∠𝐵. Third Angle Theorem: If two interior angles in one triangle are congruent to two interior angles in another triangle, then the third interior angles in the two triangles are congruent. This follows from the fact that the sum of the three interior angles in each triangle must be 180°. Example 4.1: Given 𝐴𝐷 ⊥ 𝐵𝐶 , 𝐴𝐷 bisects ∠𝐵𝐴𝐶, prove ∠𝐴𝐵𝐷 ≅ ∠𝐴𝐶𝐷. This can be proven in multiple ways. Let’s prove it with the Third Angle Theorem. Step Statement Reason 𝐴𝐷 ⊥ 𝐵𝐶. 1 Given. 𝐴𝐷 bisects ∠𝐵𝐴𝐶. ∠𝐴𝐷𝐵 is a right angle. 𝐴𝐷 ⊥ 𝐵𝐶. Perpendicular lines 2 ∠𝐴𝐷𝐶 is a right angle. form right angles. All right angles are congruent 3 ∠𝐴𝐷𝐵 ≅ ∠𝐴𝐷𝐶. (they all measure 90°). 4 ∠𝐵𝐴𝐷 ≅ ∠𝐶𝐴𝐷. 𝐴𝐷 bisects ∠𝐵𝐴𝐶. Third Angle Theorem (triangles 5 ∠𝐴𝐵𝐷 ≅ ∠𝐴𝐶𝐷 are ∆𝐴𝐷𝐵 and ∆𝐴𝐷𝐶). Version 4.2 Page 30 of 137 August 26, 2023 Chapter 4 Triangles - Basic Geometry Inequalities in Triangles Angles and their opposite sides in triangles are related. In fact, this is often reflected in the labeling of angles and sides in triangle illustrations. Angles and their opposite sides are often labeled with the same letter. An upper case letter is used for the angle and a lower case letter is used for the side. The relationship between angles and their opposite sides translates into the following triangle inequalities: If 𝒎∠𝑪 𝒎∠𝑩 𝒎∠𝑨, then 𝒄 𝑏 𝑎 If 𝒎∠𝑪 𝒎∠𝑩 𝒎∠𝑨, then 𝒄 𝒃 𝒂 That is, in any triangle,  The largest side is opposite the largest angle.  The medium side is opposite the medium angle.  The smallest side is opposite the smallest angle. Other Inequalities in Triangles Triangle Inequality: The sum of the lengths of any two sides of a triangle is greater than the length of the third side. Also, the difference of the lengths of any two sides is smaller than the length of the third side. If 𝑎 𝑏: 𝒂 𝒃 𝒄 𝒂 𝒃 and similar for the other sides. Exterior Angle Inequality: The measure of an external angle is greater than the measure of either of the two non-adjacent interior angles. That is, in the figure below: 𝒎∠𝑫𝑨𝑩 𝑚∠𝑩 and 𝒎∠𝑫𝑨𝑩 𝑚∠𝐶. Exterior Angle Equality: The measure of an external angle is equal to the sum of the measures of the two non-adjacent interior angles. That is, in the figure to the right: 𝒎∠𝑫𝑨𝑩 𝒎∠𝑩 𝒎∠𝑪. Note: the Exterior Angle Equality is typically more useful than the Exterior Angle Inequality. Version 4.2 Page 31 of 137 August 26, 2023 Chapter 4 Triangles - Basic Sides of a Triangle The lengths of the sides of a triangle are limited: given the lengths of any two sides, the length of the third side must be greater than their difference and less than their sum. That is, if the sides of a triangle have lengths 𝑎, 𝑏, and 𝑐, and you know the values of, for example, 𝑎 and 𝑏 with 𝑎 the larger of the two, then: 𝑎 𝑏 𝑐 𝑎 𝑏 Example 4.2: If a triangle has two sides with lengths 13 and 8, what are the possible lengths of the third side? If we let 𝑐 represent the length of the third side of a triangle, with 𝑎 13, 𝑏 8, then:  𝑐 must be greater than the difference of 𝑎 and 𝑏: 𝑐 13 8 → 𝑐 5.  𝑐 must be less than the sum of 𝑎 and 𝑏: 𝑐 13 8 → 𝑐 21. If we put all of this together in a single inequality, we get: 13 8 𝑐 13 8 𝟓 𝒄 𝟐𝟏 Also, as indicated above, there are limits to the lengths of sides if the measures of the interior angles of the triangle are known. In particular,  The longest side of a triangle is opposite the largest interior angle.  The shortest side of a triangle is opposite the smallest interior angle. In general, if we know that 𝑚∠𝐶 𝑚∠𝐵 𝑚∠𝐴, then we know that 𝑐 𝑏 𝑎. Example 4.3: Identify the longest segment in the diagram shown. Let’s see what we know in each of the triangles. Note that:  The sum of the angles in each triangle must be 180° and  Sides across from larger angles in the same triangle are larger. In ∆𝐴𝐵𝐶: In ∆𝐴𝐷𝐸: In ∆𝐴𝐶𝐷:  𝑚∠𝐵𝐴𝐶 43°  𝑚∠𝐸𝐴𝐷 38°  𝐴𝐵 𝐵𝐶 𝐴𝐶  𝐷𝐸 𝐴𝐸 𝐴𝐷  𝐶𝐷 𝐴𝐶 𝐴𝐷 Therefore, the two candidates for longest segment are 𝐴𝐶 and 𝐴𝐷. Looking closer at the above inequalities, we notice that in ∆𝐴𝐶𝐷, we have 𝐴𝐶 𝐴𝐷. Therefore, the longest segment is: 𝑨𝑫. Version 4.2 Page 32 of 137 August 26, 2023 Chapter 4 Triangles - Basic The discussion above addresses angles within a single triangle. There is another relationship that allows us to compare the lengths of sides in two different triangles. In particular, If two triangles have two pairs of congruent sides, consider the angles between the congruent sides. The triangle with the larger of these angles has the larger side opposite that angle. This is illustrated in the next example. Example 4.4: Find the range of values for 𝑥. Note: never trust the relative sizes of angles and sides in a diagram. For example, the two sides with length 9 in this diagram are drawn with different lengths! We know two things involving 𝑥:  The side labeled 3𝑥 4 must be positive. So, 3𝑥 4 0.  The two angles shown (39°) and (41°) share two congruent sides (one side with length 9 and one side of unknown length that is shared by the two angles). Therefore, the side opposite the smaller angle must be smaller than the side opposite the larger angle. So, 3𝑥 4 17. Combining these two inequalities into a single compound inequality, and solving: Starting inequality: 0 3𝑥 4 17 Add 4: 4 3𝑥 21 𝟒 Divide by 3: 𝒙 𝟕 𝟑 Example 4.5: Given ∆ABC with A 3, 4 , B 7, 1 , C 2, 1 , and median AD, find the coordinates of point D. Many times, you need to draw the situation for a given problem. This is not one of those times. Point D is the midpoint of the side of the triangle opposite the given vertex. In this problem, Point A is the vertex in question (it is on the median 𝐴𝐷). So, Point D is the midpoint of the points B 7, 1 and C 2, 1. So, the coordinates of Point D are: 7, 1 2, 1 2 𝟒. 𝟓, 𝟎 Version 4.2 Page 33 of 137 August 26, 2023 Chapter 4 Triangles - Basic Example 4.6: Given ∆ABC with A 2,5 , B 3,5 , C 6, 1 , and altitude CD, find the coordinates of point D. An altitude of a triangle is a line segment drawn from a vertex to a point on the opposite side (extended, if necessary) that is perpendicular to that side. This problem is very straightforward once you graph it. To find the base point of the altitude, we can look at the intersection of the two lines on which Point D lies. Line containing 𝐵𝐴: 𝑦 5 Line containing 𝐶𝐷. 𝑥 6 is perpendicular to 𝑦 5 and contains C 6, 1. Therefore, Point D has coordinates: 𝟔, 𝟓. Example 4.7: Write and solve an inequality for 𝑥. Each side must have a positive measure, so: 𝑥 2 0 𝑥 2 Also, in the triangle on the left, we have: 7 6 𝑥 2 7 6 1 𝑥 2 13 3 𝑥 15 Next, both outside triangles have sides of length 6 and 7 with angles between them. Since the measure of the angle in the triangle on the left 54° is less than the one in the triangle on the right 67° , the opposite side on the left must be less than the opposite side on the right. So, 𝑥 2 11. 𝑥 13 Putting it all together, we have: 3 𝑥, equivalent to 𝑥 3, which is more restrictive than 𝑥 2, so we use the more restrictive 3 𝑥. We also have: 𝑥 13, which is more restrictive than 𝑥 15, so we use the more restrictive 𝑥 13. Finally, since 3 𝑥 and 𝑥 13, we have 𝟑 𝒙 𝟏𝟑 Version 4.2 Page 34 of 137 August 26, 2023 Chapter 4 Triangles - Basic Geometry Types of Triangles Scalene Isosceles A Scalene Triangle has 3 sides of different An Isosceles Triangle has 2 sides the same lengths. Because the sides are of length (i.e., congruent). Because two different lengths, the angles must also be sides are congruent, two angles must also of different measures. be congruent. Equilateral Right An Equilateral Triangle has all 3 sides the A Right Triangle is one that contains a 90⁰ same length (i.e., congruent). Because all angle. It may be scalene or isosceles, but 3 sides are congruent, all 3 angles must cannot be equilateral. Right triangles also be congruent. This requires each have sides that meet the requirements of angle to be 60⁰. the Pythagorean Theorem. 60⁰ 60⁰ 60⁰ Version 4.2 Page 35 of 137 August 26, 2023 Chapter 4 Triangles - Basic Example 4.8: Find the values of 𝑥 and 𝑦 based on the diagram. This problem becomes easier if we label a few more angles. See the diagram on the right. Angles opposite congruent sides in isosceles triangles are congruent, which helps with our labeling. In the triangle on the right, the sum of the interior angles must be 180°, so, 𝑏 180 37 37 106. The adjacent angles marked 𝑎° and 𝑏° form a linear pair, so, 𝑎 180 106 74. The center triangle has two angles of 𝑎° and one angle of 𝑦°, which must add to 180°, so, 𝒚 180 74 74 𝟑𝟐. Finally, along the top right, angles marked 37°, 𝑎°, and 𝑥° must add to 180° in order to form a straight angle, so, 𝒙 180 37 74 𝟔𝟗. Example 4.9: Find the value of 𝑦 and the perimeter of the triangle. Legs opposite congruent angles in isosceles triangles are congruent. 𝑦 5𝑦 24 𝑦 5𝑦 24 0 𝑦 8 𝑦 3 0 𝒚 𝟖, 𝟑 (2 possibilities) If we plug each of these values into the lengths of the sides shown in the diagram, we always get positive numbers, so there are two cases. If we had gotten a length that was negative for either 𝑦 8 or 𝑦 3, we would have had to discard that solution. The perimeter of the triangle is: 𝑃 𝑦 4𝑦 15 5𝑦 24 𝑦 9𝑦 39. Case 1 (𝑦 8): 𝑃 𝑦 9𝑦 39 8 9∙8 39 𝟏𝟕𝟓. (we are not given units) Sides of this triangle are 64, 64, 47, which gives a viable triangle. Case 2 (𝑦 3): 𝑃 𝑦 9𝑦 39 3 9∙ 3 39 𝟐𝟏. Sides of this triangle are 9, 9, 3, which gives a viable triangle. Version 4.2 Page 36 of 137 August 26, 2023 Chapter 4 Triangles - Basic Geometry Congruent Triangles The following theorems present conditions under which triangles are congruent. Side-Angle-Side (SAS) Congruence SAS congruence requires the congruence of two sides and the angle between those sides. Note that there is no such thing as SSA congruence; the congruent angle must be between the two congruent sides. ide-Side-Side (SSS) Congruence SSS congruence requires the congruence of all three sides. If all of the sides are congruent then all of the angles must be congruent. The converse is not true; there is no such thing as AAA congruence. Angle-Side-Angle (ASA) Congruence ASA congruence requires the congruence of two angles and the side between those angles. Note: ASA and AAS combine to provide congruence of two triangles whenever any two angles and any one side of the Angle-Angle-Side (AAS) Congruence triangles are congruent. AAS congruence requires the congruence of two angles and a side which is not between those angles. Hypotenuse Leg (HL) Congruence HL can be used if the triangles in question have right angles. It requires the congruence of the hypotenuse and one of the other legs. Version 4.2 Page 37 of 137 August 26, 2023 Chapter 4 Triangles - Basic CPCTC CPCTC means “corresponding parts of congruent triangles are congruent.” It is a very powerful tool in geometry proofs and is often used shortly after a step in the proof where a pair of triangles is proved to be congruent. Example 4.10: Given that BE is a perpendicular bisector of CD, find ED. In the diagram, CA ≅ DA because BE bisects CD. So, ∆CAB ≅ ∆DAB by SAS, and ∆CAE ≅ ∆DAE by SAS. The two hypotenuses (yep, that’s the plural form of hypotenuse) of the triangles on the right side of the diagram are congruent. So, 7x 10 2x 20 5x 30 x 6 CA DA, so y 4 Finally, ED EC x 2y (because ∆CAE ≅ ∆DAE, and ED and EC are corresponding parts of those congruent triangles). ED EC 6 2 4 𝟏𝟒 Example 4.11: Given ∆𝑃𝑄𝑅 ≅ ∆𝐽𝐾𝐿, 𝑃𝑄 9𝑥 45, 𝐽𝐾 6𝑥 15, 𝐾𝐿 2𝑥, 𝐽𝐿 5𝑥, what is the value of 𝑥? It’s helpful to draw a picture for this problem. Notice that congruent segments 𝑃𝑄 and 𝐽𝐾 have measures 9𝑥 45 and 6𝑥 15. Then: 9𝑥 45 6𝑥 15 3𝑥 60 𝒙 𝟐𝟎 We are not quite finished, even though we found a value for 𝑥. We need to check the sides of ∆𝐽𝐾𝐿 to make sure this results in a viable triangle: 2𝑥 40, 5𝑥 100, 6𝑥 15 135 Sides of 40, 100, 135 are viable in a triangle because 40 100 135. Note that if 𝑃𝑄 12𝑥 45, we would have calculated 𝑥 10. Then, the sides would have been 20, 50, 75, which is not a viable triangle because 20 50 75. If this were the case, this problem would have no solution. Version 4.2 Page 38 of 137 August 26, 2023 Chapter 4 Triangles - Basic Example 4.12: Given 𝐴𝐷 ⊥ 𝐵𝐶 , 𝐴𝐷 bisects ∠𝐵𝐴𝐶, prove ∠𝐵 ≅ ∠𝐶. It looks like we want to head toward ∆𝐴𝐷𝐵 ≅ ∆𝐴𝐷𝐶, and use CPCTC. Step Statement Reason 𝐴𝐷 ⊥ 𝐵𝐶. 1 Given. 𝐴𝐷 bisects ∠𝐵𝐴𝐶. ∠𝐴𝐷𝐵 is a right angle. 𝐴𝐷 ⊥ 𝐵𝐶. Perpendicular lines 2 ∠𝐴𝐷𝐶 is a right angle. form right angles. 3 ∠𝐴𝐷𝐵 ≅ ∠𝐴𝐷𝐶. All right angles are congruent. 4 𝐴𝐷 ≅ 𝐴𝐷. Reflexive property of congruence. 5 ∠𝐵𝐴𝐷 ≅ ∠𝐶𝐴𝐷. 𝐴𝐷 bisects ∠𝐵𝐴𝐶. 6 ∆𝐴𝐷𝐵 ≅ ∆𝐴𝐷𝐶 ASA congruence theorem. 7 ∠𝐵 ≅ ∠𝐶 CPCTC. Example 4.13: Given 𝐴𝐷 ∥ 𝐶𝐵 , 𝐴𝐵 ∥ 𝐶𝐷 , prove ∠𝐵 ≅ ∠𝐷 With parallel lines, we will typically look for alternate interior angles or corresponding angles to prove things. Also, this looks like a situation where we prove congruent triangles and can use CPCTC. Step Statement Reason 𝐴𝐷 ∥ 𝐶𝐵. 1 Given. 𝐴𝐵 ∥ 𝐶𝐷. ∠𝐵𝐴𝐶 ≅ ∠𝐷𝐶𝐴. Alternate interior angles of 𝐴𝐵 ∥ 𝐶𝐷 , 2 with 𝐴𝐶 a transversal. 3 ∠𝐵𝐶𝐴 ≅ ∠𝐷𝐴𝐶. Alternate interior angles of 𝐴𝐷 ∥ 𝐶𝐵 , with 𝐴𝐶 a transversal. 4 𝐴𝐶 ≅ 𝐴𝐶. Reflexive property of congruence. 5 ∆𝐵𝐴𝐶 ≅ ∆𝐷𝐶𝐴 ASA congruence theorem. 6 ∠𝐵 ≅ ∠𝐷 CPCTC. Version 4.2 Page 39 of 137 August 26, 2023 Chapter 4 Triangles - Basic Geometry Centers of Triangles The following are all points which can be considered the center of a triangle. Centroid (Medians) The centroid is the intersection of the three medians of a triangle. A median is a line segment drawn from a vertex to the midpoint of the side of the triangle that is opposite the vertex.  The centroid is located 2/3 of the way from a vertex to the opposite side. That is, the distance from a vertex to the centroid is double the length from the centroid to the midpoint of the opposite line.  The medians of a triangle create 6 inner triangles of equal area. Orthocenter (Altitudes) The orthocenter is the intersection of the three altitudes of a triangle. An altitude is a line segment drawn from a vertex to a point on the opposite side (extended, if necessary) that is perpendicular to that side.  In an acute triangle, the orthocenter is inside the triangle.  In a right triangle, the orthocenter is the right angle vertex.  In an obtuse triangle, the orthocenter is outside the triangle. Circumcenter (Perpendicular Bisectors) The circumcenter is the intersection of the perpendicular bisectors of the three sides of the triangle. A perpendicular bisector is a line which Euler Line: Interestingly, the centroid, orthocenter both bisects the side and is perpendicular to the and circumcenter of a side. The circumcenter is also the center of the triangle are collinear (i.e., circle circumscribed about the triangle. lie on the same line, which is called the Euler  In an acute triangle, the circumcenter is inside the triangle. Line).  In a right triangle, the circumcenter is the midpoint of the hypotenuse.  In an obtuse triangle, the circumcenter is outside the triangle. Incenter (Angle Bisectors) The incenter is the intersection of the angle bisectors of the three angles of the triangle. An angle bisector cuts an angle into two congruent angles, each of which is half the measure of the original angle. The incenter is also the center of the circle inscribed in the triangle. Version 4.2 Page 40 of 137 August 26, 2023 Chapter 4 Triangles - Basic Example 4.14: Given ∆CAB, CG 3𝑥 2, GF 𝑥 3, find 𝑥 and 𝐶𝐹. Centroid  The centroid is the intersection of the three medians of a triangle.  A median is a line segment drawn from a vertex to the midpoint of the side of the triangle that is opposite the vertex.  The centroid is located 2/3 of the way from a vertex to the opposite side.  The medians of a triangle create 6 inner triangles of equal area. From the diagram, we can see that Points D, E, F are midpoints of the sides of ∆ABC. So, AD, BE, CF are medians of ∆ABC. Point G is the centroid of ∆ABC because it is the intersection of the three medians of the triangle. Therefore, CG 2 GF 3𝑥 2 2 𝑥 3 3𝑥 2 2𝑥 6 𝒙 𝟖 Then, 𝐂𝐅 CG GF 3𝑥 2 𝑥 3 4𝑥 1 4 8 1 𝟑𝟑 Version 4.2 Page 41 of 137 August 26, 2023 Chapter 4 Triangles - Basic Geometry Length of Altitude, Median and Angle Bisector Altitude (Height) The formula for the length of a height of a triangle is derived from Heron’s formula for the area of a triangle: 𝟐 𝒔 𝒔 𝒂 𝒔 𝒃 𝒔 𝒄 𝒉 𝒄 𝟏 where, 𝒔 𝒂 𝒃 𝒄 , and 𝟐 𝒂, 𝒃, 𝒄 are the lengths of the sides of the triangle. Median The formula for the length of a median of a triangle is: 𝟏 𝒎 𝟐𝒂𝟐 𝟐𝒃𝟐 𝒄𝟐 𝟐 where, 𝒂, 𝒃, 𝒄 are the lengths of the sides of the triangle. Angle Bisector The formula for the length of an angle bisector of a triangle is: 𝒄𝟐 𝒕 𝒂𝒃 𝟏 𝟐 𝒂 𝒃 where, 𝒂, 𝒃, 𝒄 are the lengths of the sides of the triangle. Example 4.15: Find the length of CF, if CF is a median of ∆ABC. Point F bisects AB, so AB 2∙5 10. From the formula above, we have: 1 𝐂𝐅 2 ∙ AC 2 ∙ CB AB 2 1 1 1 2∙4 2∙8 10 √60 ∙ 2√15 √𝟏𝟓 2 2 2 Version 4.2 Page 42 of 137 August 26, 2023 Chapter 5 Polygons Geometry Polygons - Basics Basic Definitions Polygon: a closed path of three or more line segments, where:  no two sides with a common endpoint are collinear, and  each segment is connected at its endpoints to exactly two other segments. Side: a segment that is connected to other segments (which are also sides) to form a polygon. Vertex: a point at the intersection of two sides of the polygon. (plural form: vertices) Diagonal: a segment, from one vertex to another, which is not a side. Vertex Diagonal Side Concave: A polygon in which it is possible to draw a diagonal “outside” the polygon. (Notice the orange diagonal drawn outside the polygon at right.) Concave polygons actually look like they have a “cave” in them. Convex: A polygon in which it is not possible to draw a diagonal “outside” the polygon. (Notice that all of the orange diagonals are inside the polygon at right.) Convex polygons appear more “rounded” and do not contain “caves.” Names of Some Common Polygons Number Number Names of polygons of Sides Name of Polygon of Sides Name of Polygon are generally formed 3 Triangle 9 Nonagon from the Greek language; however, 4 Quadrilateral 10 Decagon some hybrid forms of 5 Pentagon 11 Undecagon Latin and Greek (e.g., 6 Hexagon 12 Dodecagon undecagon) have 7 Heptagon 20 Icosagon crept into common usage. 8 Octagon n n‐gon Version 4.2 Page 43 of 137 August 26, 2023 Chapter 5 Polygons Geometry Polygons – More Definitions Definitions “Advanced” Definitions: Equilateral: a polygon in which all of the sides are equal in length. Simple Polygon: a Equiangular: a polygon in which all of the angles have the same polygon whose sides do measure. not intersect at any location other than its Regular: a polygon which is both equilateral and equiangular. That endpoints. Simple is, a regular polygon is one in which all of the sides have the same polygons always divide a length and all of the angles have the same measure. plane into two regions – one inside the polygon and one outside the polygon. Interior Angle: An angle formed by two sides of a polygon. The Complex Polygon: a angle is inside the polygon. polygon with sides that Exterior Angle: An angle formed by one side of a polygon and the intersect someplace other line containing an adjacent side of the polygon. The angle is outside than their endpoints (i.e., the polygon. not a simple polygon). Complex polygons do not always have well-defined insides and outsides. Exterior Interior Skew Polygon: a polygon Angle Angle for which not all of its vertices lie on the same plane. How Many Diagonals Does a Convex Polygon Have? Believe it or not, this is a common question with a simple solution. Consider a polygon with 𝒏 𝟑 sides and, therefore, 𝒏 vertices.  Each of the n vertices of the polygon can be connected to 𝒏 𝟑 other vertices with diagonals. That is, it can be connected to all other vertices except itself and the two to which it is connected by sides. So, there are 𝒏 ∙ 𝒏 𝟑 lines to be drawn as diagonals.  However, when we do this, we draw each diagonal twice because we draw it once from each of its two endpoints. So, the number of diagonals is actually half of the number we calculated above.  Therefore, the number of diagonals in an n-sided polygon is: 𝑛∙ 𝑛 3 2 Version 4.2 Page 44 of 137 August 26, 2023 Chapter 5 Polygons Geometry Interior and Exterior Angles of a Polygon Interior Angles Interior Angles The sum of the interior angles in an 𝑛-sided polygon is: Sum of Each Sides Interior Interior ∑ 𝑛 2 ∙ 180° Angles Angle 3 180⁰ 60⁰ If the polygon is regular, you can calculate the measure of 4 360⁰ 90⁰ each interior angle as: 5 540⁰ 108⁰ 6 720⁰ 120⁰ ∙ ° 7 900⁰ 129⁰ 8 1,080⁰ 135⁰ 9 1,260⁰ 140⁰ Notation: The Greek letter “Σ” is equivalent 10 1,440⁰ 144⁰ to the English letter “S” and is math short-hand for a summation (i.e., addition) of things. Exterior Angles Exterior Angles No matter how many sides there are in a polygon, the sum Sum of Each of the exterior angles is: Sides Exterior Exterior Angles Angle ∑ 360⁰ 3 360⁰ 120⁰ 4 360⁰ 90⁰ If the polygon is regular, you can calculate the measure of 5 360⁰ 72⁰ each exterior angle as: 6 360⁰ 60⁰ 7 360⁰ 51⁰ ⁰ 8 360⁰ 45⁰ 9 360⁰ 40⁰ 10 360⁰ 36⁰ Version 4.2 Page 45 of 137 August 26, 2023 Chapter 6 Quadrilaterals Geometry Definitions of Quadrilaterals Name Definition Quadrilateral A polygon with 4 sides. A quadrilateral with two consecutive pairs of congruent sides, but Kite with opposite sides not congruent. Trapezoid A quadrilateral with exactly one pair of parallel sides. Isosceles Trapezoid A trapezoid with congruent legs. Parallelogram A quadrilateral with both pairs of opposite sides parallel. Rectangle A parallelogram with all angles congruent (i.e., right angles). Rhombus A parallelogram with all sides congruent. Square A quadrilateral with all sides congruent and all angles congruent. Quadrilateral Tree: Quadrilateral Kite Parallelogram Trapezoid Rectangle Rhombus Isosceles Trapezoid Square Version 4.2 Page 46 of 137 August 26, 2023 Chapter 6 Quadrilaterals Geometry Figures of Quadrilaterals Kite Trapezoid Isosceles Trapezoid  2 consecutive pairs of  1 pair of parallel sides  1 pair of parallel sides congruent sides (called “bases”)  Congruent legs  1 pair of congruent  Angles on the same  2 pair of congruent base opposite angles “side” of the bases are angles  Diagonals perpendicular supplementary  Diagonals congruent Parallelogram Rectangle  Both pairs of opposite sides parallel  Parallelogram with all angles  Both pairs of opposite sides congruent congruent (i.e., right angles)  Both pairs of opposite angles congruent  Diagonals congruent  Consecutive angles supplementary  Diagonals bisect each other Rhombus Square  Parallelogram with all sides congruent  Both a Rhombus and a Rectangle  Diagonals perpendicular  All angles congruent (i.e., right angles)  Each diagonal bisects a pair of  All sides congruent opposite angles Version 4.2 Page 47 of 137