Let's Review Regents: Geometry 2020 PDF

Let’s Review Regents: Geometry 2020 Andre Castagna, Ph.D. Mathematics Teacher Albany High School Albany, New York Table of Contents Cover Title Page Copyright Information Dedication Preface Chapter 1: The Tools of Geometry 1.1 The Building Blocks of Geometry 1.2 Basic Relationships Among Points, Lines, and Planes 1.3 Brief Review of Algebra Skills Chapter 2: Angle and Segment Relationships 2.1 Basic Angle Relationships 2.2 Bisectors, Midpoint, and the Addition Postulate 2.3 Angles in Polygons 2.4 Parallel Lines 2.5 Angles and Sides in Triangles Chapter 3: Constructions 3.1 Basic Constructions 3.2 Constructions that Build on the Basic Constructions 3.3 Points of Concurrency, Inscribed Figures, and Circumscribed Figures Chapter 4: Introduction to Proofs 4.1 Structure and Strategy of Writing Proofs 4.2 Using Key Idea Midpoints, Bisectors, and Perpendicular Lines 4.3 Properties of Equality 4.4 Using Vertical Angles, Linear Pairs, and Complementary and Supplementary Angles 4.5 Using Parallel Lines 4.6 Using Triangle Relationships Chapter 5: Transformations and Congruence 5.1 Rigid Motion and Similarity Transformations 5.2 Properties of Transformations 5.3 Transformations in the Coordinate Plane 5.4 Symmetry 5.5 Compositions of Rigid Motions Chapter 6: Triangle Congruence 6.1 The Triangle Congruence Criterion 6.2 Proving Triangles Congruent 6.3 CPCTC 6.4 Proving Congruence by Transformations Chapter 7: Geometry in the Coordinate Plane 7.1 Length, Distance, and Midpoint 7.2 Perimeter and Area Using Coordinates 7.3 Slope and Equations of Lines 7.4 Equations of Parallel and Perpendicular Lines 7.5 Equations of Lines and Transformations 7.6 The Circle Chapter 8: Similar Figures and Trigonometry 8.1 Similar Figures 8.2 Proving Triangles Similar and Similarity Transformations 8.3 Similar Triangle Relationships 8.4 Right Triangle Trigonometry Chapter 9: Parallelograms and Trapezoids 9.1 Parallelograms 9.2 Proofs with Parallelograms 9.3 Properties of Special Parallelograms 9.4 Trapezoids 9.5 Classifying Quadrilaterals and Proofs Involving Special Quadrilaterals 9.6 Parallelograms and Transformations Chapter 10: Coordinate Geometry Proofs 10.1 Tools and Strategies of Coordinate Geometry Proofs 10.2 Parallelogram Proofs 10.3 Triangle Proofs Chapter 11: Circles 11.1 Definitions, Arcs, and Angles in Circles 11.2 Congruent, Parallel, and Perpendicular Chords 11.3 Tangents 11.4 Angle-Arc Relationships with Chords, Tangents, and Secants 11.5 Segment Relationships in Intersecting Chords, Tangents, and Secants 11.6 Area, Circumference, and Arc Length Chapter 12: Solids and Modeling 12.1 Prisms and Cylinders 12.2 Cones, Pyramids, and Spheres 12.3 Cross Sections and Solids of Revolution 12.4 Proving Volume and Area by Dissection, Limits, and Cavalieri’s Principle 12.5 Modeling and Design Answers and Solutions to Practice Exercises Glossary of Geometry Terms Summary of Geometric Relationships and Formulas About the Exam Appendix I: The Common Core Geometry Learning Standards Appendix II: Practice and Test-Taking Tips June 2018 Regents Exam © Copyright 2018, 2017, 2016 by Kaplan, Inc., d/b/a Barron’s Educational Series All rights reserved under International and Pan-American Copyright Conventions. By payment of the required fees, you have been granted the non-exclusive, non-transferable right to access and read the text of this eBook on screen. No part of this text may be reproduced, transmitted, downloaded, decompiled, reverse engineered, or stored in or introduced into any information storage and retrieval system, in any form or by any means, whether electronic or mechanical, now known or hereina er invented, without the express written permission of the publisher. Published by Kaplan, Inc., d/b/a Barron’s Educational Series 750 Third Avenue New York, NY 10017 www.barronseduc.com ISBN: 978-1-5062-7081-4 10 9 8 7 6 5 4 3 2 1 Dedication To my loving wife Loretta, who helped make this endeavor possible with her unwavering support; my geometry buddy, Eva; and my future geometry buddies, Rose and Henry. PREFACE This book presents the concepts, applications, and skills necessary for students to master the Geometry Common Core curriculum. Topics are grouped and presented in an easy-to-understand manner similar to what might be encountered in the classroom. Both students preparing for the Regents exam and teachers planning daily classroom lessons will find this book a valuable resource. SPECIAL FEATURES OF THIS BOOK Aligned with the Common Core This book has been rewritten to reflect the curriculum changes found in the Common Core—both in content and in degree of critical thinking expected. The Geometry Common Core curriculum has placed more emphasis on transformational geometry, especially as applied to congruence. This emphasis has been integrated throughout the book in each chapter. Example problems and practice exercises can be found throughout this book. These demonstrate higher-level thinking, applying multiple concepts in a single problem, making connections between concepts, and demonstrating understanding in words. Easy-to-Read Format Topics are arranged in a logical manner so that examples and practice problems build on material from previous chapters. This format complements the presentation of material a student may see in the classroom. The format makes this book an excellent resource both for improving understanding throughout the school year and for preparing for the Regents exam. Numerous example problems with step-by-step solutions are provided, along with many detailed figures and diagrams to illustrate and clarify the topic at hand. Each subsection begins with a “Key Ideas” summary of the major facts the student should take away from that section. “Math Facts” can be found throughout the book and provide further insight or interesting historical notes. Review of Algebra Skills The first chapter of this book contains a review of practice problems of the algebra skills required to work through some of the problems encountered in geometry. Although many geometry problems involve pure reasoning, logic, and application of geometric principles, students can expect to encounter numerical problems in which they must apply equation-solving skills learned in previous years. An Introduction to Proofs A step-by-step guide to writing accurate geometry proofs can be found in chapter 4. Students are provided with the opportunity to develop the skills needed to write proofs by starting with just a small handful of geometric concepts and tools. The scaﬀolding provided in this chapter will help students develop the skills and confidence needed to meet the demands of the Common Core curriculum successfully. Both the two-column and paragraph formats of proofs are presented. A Wide Variety of Practice Problems and Two Actual Regents Exams The practice problems at the end of each subsection feature a range of complexity. Basic application of skills, including applying a formula or recalling a definition, lead into multistep problem solving and critical analysis. Problems that ask students to put their understanding in writing can also be found in each chapter. These practice problems include a large number of multiple-choice questions, similar to what can be expected on the Regents exam. The answers to all “Check Your Understanding” problems are provided. Two actual Regents exams with answer keys are included. These give students valuable experience with the style, format, and length of the Geometry Common Core Regents exam. A Detailed Description of the Exam Format and Study Tips The format of the exam, point distribution among topics, and scoring conversion are thoroughly explained. Students can use this information to help focus their eﬀorts and ensure that they thoroughly master topics with high point value. Study tips and advice for test day are provided to help students make the most of their study time. The Common Core Standards A complete list of the Geometry Common Core standards can be found in Appendix I. All teachers should be thoroughly familiar with the content of these standards. WHAT’S NOT IN THIS BOOK This review book does not provide proofs for all the theorems found within it. In fact, it shows fewer proofs than the typical geometry textbook. The theorems and proofs that were included in this work are those that: illustrate the level of complexity expected of students demonstrate specific strategies and approaches that a student may be expected to apply are specifically required within the Common Core geometry learning standards Students are strongly encouraged to read and understand the proofs that are included here carefully. They should be able to complete on their own any proof noted to be specifically required by the Core Curriculum. Of course, rote memorization of these proofs is strongly discouraged. Instead, students should familiarize themselves with the tools and strategies used in proofs and then be able to work through the proofs by applying their critical thinking and geometry skills. WHO WILL BENEFIT FROM USING THIS BOOK Students who want to achieve their best-possible grade in the classroom and on the Regents exam will benefit. Students may use this book as a study guide for both their day-to-day lessons and for the Regents exam. Teachers who would like an additional resource when planning geometry lessons aligned to the Common Core will benefit. Curriculum and district administrators who want to ensure their math department’s curriculum is aligned to the Common Core will benefit. Chapter THE TOOLS One OF GEOMETRY 1.1 THE BUILDING BLOCKS OF GEOMETRY K I The building blocks of geometry are the point, line, and plane. The definitions of the other geometric figures can all be traced back to these three. We can think of the point, line, and plane as analogous to the elements in chemistry. All compounds are built up from the elements in the same way that the geometric figures are built up of points, lines, and planes. Along with definitions, we also look at the notation used for each. The ability to interpret vocabulary and notation is important for success in geometry. POINT, LINES, AND PLANES A point is location in space. It is zero dimensional, having no length, width, or thickness. Points are represented by a dot and named with a capital letter, as shown by Point A in Figure 1.1a. Don’t let the dot confuse you—points are infinitely small. Even the smallest dot you can draw is two-dimensional. Figure 1.1 Points, lines, and planes A line is a set of points extending without end in opposite directions. Lines can be curved or straight. In this book, we will use the term line to refer to straight lines. Lines are one dimensional. They have an infinite length but have no height or thickness. They are represented by a double arrow to indicate the infinite length. They are named with any two points on the line as shown in Figure 1.1b or with a lowercase letter as shown in Figure 1.1c. Three or more points may also be used if we want to indicate the line continues straight through multiple points as in Figure 1.1d. A plane is a set of points that forms a flat surface. Planes are two- dimensional. They have infinite length and width but no height. A tabletop or wall can represent a portion of a plane. Remember, though, that the plane continues infinitely beyond the boundaries of the tabletop or wall in each direction. Planes are named with any three points that do not lie on the same line, as shown in Figure 1.1e, or with a capital letter, as shown in Figure 1.1f. Example 1 Name the following line in 7 diﬀerent ways. Solution: Example 2 Name the plane in two diﬀerent ways. Solution: Plane QRS, plane Z Example 3 How many points lie on Solution: An infinite number. Every line contains an infinite number of points. We just show a few of them when representing and naming a line. RAYS AND SEGMENTS A ray is a portion of a line that has one endpoint and continues infinitely in one direction. A ray is named by the endpoint followed by any other point on the ray. When naming a ray, an arrow is used. The endpoint of the arrow is over the endpoint of the ray. Figure 1.2 illustrates ray with endpoint A and ray with endpoint B. Figure 1.2 Rays and When two rays share an endpoint and form a straight line, the rays are called opposite rays. We say the union of the two rays forms a straight line. A line segment is a portion of a line with two endpoints. It is named using the two endpoints in either order with an overbar. Figure 1.3 illustrates segment or. The length of a segment is the distance between the two endpoints. The length of can be referred to as FG or |FG|. In some situations, we may wish to specify a particular starting point and ending point for the segment by using a directed segment. For example, a person walking along directed segment FG would begin at point F and walk directly to point G. Figure 1.3 Segment or Remember that an infinite number of points are on any line, ray, or segment even though they are not explicitly shown in a figure. Also remember that lines, rays, and segments can be considered to exist even though they are not explicitly shown in a figure. Example 1 Name each segment and ray in the figure. Solution: Segments and , rays and Example 2 If has a length of 5, what is the length of ? Solution: also has a length of 5 because and are the same segment. ANGLES An angle is the union of two rays with a common endpoint. The common endpoint is called the vertex. Angles can be named using three points—a point on the first ray, the vertex, and a point on the second ray. The vertex is always listed in the middle. Alternatively, one can use only the vertex point or a reference number. Figure 1.4 shows the diﬀerent ways to name an angle. Figure 1.4 Naming angles Angles are measured in degrees. One degree is defined as of the way around a circle. Halfway around the circle is 180°, and one-quarter around is 90°. The measure of an angle can be specified using the letter m. For example, m∠RST = 30°. Angles can be classified by their degree measure. Acute angle—an angle whose measure is less than 90°. Right angle—an angle whose measure is exactly 90°. Obtuse angle—an angle whose measure is more than 90° and less than 180°. Straight angle—an angle whose measure is exactly 180°. Figure 1.5 shows examples of each type of angle. The square positioned at the vertex of the right angle is o en used to specify a right angle. Figure 1.5 Classification of angles M F Our definition of the degree as of a rotation around a center point has been used since ancient times. No one knows for sure why was chosen. One theory is that it originated with ancient Babylonian mathematicians, who used a base-60 number system instead of the base-10 system we use today. They divided a circle into 6 congruent equilateral triangles with 60° central angles. Then the ancient Babylonians subdivided each central angle into 60 parts. Another theory is that the circle was divided into 360 parts because one year is approximately 360 days. Either way, 360 is a convenient number to partition the circle with because 360 is divisible by 1, 2, 3, 4, 5, 6, 8, 9, and 10. Example 1 Name one angle and two rays. Solution: , , Example 2 Name each angle in 3 ways, and classify each angle. Solution: ∠R, ∠SRT, ∠TRS; acute angle ∠E, ∠DEF, ∠FED; obtuse angle ∠I, ∠HIJ, ∠JIH; right angle ADJACENT ANGLES Angles that share a common ray and vertex but no interior points are adjacent angles. In Figure 1.6, ∠ABC and ∠CBD are adjacent angles. ∠ABC and ∠ABD are not to be considered adjacent because they share interior points in the region of ∠CBD. To avoid confusion, always use three vertices or a reference number when naming adjacent angles. Using the vertex alone would be ambiguous. Figure 1.6 Adjacent angles ∠ABC and ∠CBD Example 1 Name 3 pairs of adjacent angles. Solution: ∠AOB and ∠BOC, ∠BOC and ∠COA, ∠COA and ∠AOB POLYGONS A polygon is a closed figure with straight sides. They are named for the number of sides. Be familiar with these common polygons. Triangle—3 sides Quadrilateral—4 sides Pentagon—5 sides Hexagon—6 sides Octagon—8 sides Decagon—10 sides The intersection of two sides in a polygon is called a vertex (plural is vertices). The vertices are used to name specific polygons by listing the vertices in order around the polygon. They can be called out either clockwise or counterclockwise but must always be stated in continuous order—no skipping allowed. Figure 1.7 shows some polygons with their names. We can list the vertices of the triangle in any order since it would be impossible to skip a vertex. For the quadrilateral, the name EFGD is valid but EFDG is not. For triangles, we o en precede the vertices with the triangle symbol, ∆, so triangle ABC would be referred to as ∆ABC. Figure 1.7 Triangle ABC (or ΔABC), quadrilateral DEFG, pentagon HIJKL When all the sides of a polygon are congruent to one another (equilateral) and all the angles of the polygon are congruent to one another (equiangular), we refer to that polygon as regular. So a square is an example of a regular quadrilateral, while a rectangle may have two sides with lengths diﬀerent from the other two. Example Sketch hexagon RSTUVW. Name each side and each angle. Solution: Sides Angles ∠R, ∠S, ∠T, ∠U, ∠V, ∠W CLASSIFYING TRIANGLES Triangles can be classified by their angle lengths and measures as shown in Figure 1.8 below. Figure 1.8 Triangle classifications Classifying triangles by sides: Scalene—no congruent sides, no congruent angles Isosceles—at least two congruent sides, two congruent angles Equilateral—three congruent sides, three congruent angles Classifying triangles by angles: Acute—all angles are acute Right—one right angle Obtuse—one obtuse angle Example 1 ∆ABC has side lengths AB = 2, BC = 1, and AC = 1.7. ∠A measures 30°, ∠B measures 60°, and ∠C measures 90°. Classify the triangle. Solution: ∆ABC is a right acute triangle. Four special segments can be drawn in a triangle, and every triangle has three of each. These special segments are the altitude, median, angle bisector, and perpendicular bisector, shown in Figure 1.9. Figure 1.9 Special segments in triangles Altitude—a segment from a vertex perpendicular to the opposite side Median—a segment from a vertex to the midpoint of the opposite side Angle bisector—a line, segment, or ray passing through the vertex of a triangle and bisecting that angle Perpendicular bisector—a segment, line, or ray that is perpendicular to and passes through the midpoint of a side Example 2 In ∆ABC, is drawn such that D lies on and is perpendicular to. What special segment is ? Solution: is an altitude. It has an endpoint at a vertex and is perpendicular to the opposite side of the triangle. Example 3 In triangle ∆ABC, is drawn such that ∠BAD and ∠CAD have the same measure. What special segment is ? Solution: is an angle bisector of ∆ABC. Check Your Understanding of Section 1.1 A. Multiple-Choice 1. In ∆FGH, K is the midpoint of. What type of segment is ? (1) median (2) altitude (3) angle bisector (4) perpendicular bisector 2. The side lengths of a triangle are 8, 10, and 12. The triangle can be classified as (1) equilateral (2) isosceles (3) scalene (4) right 3. The side lengths of a triangle are 12, 12, and 15. The triangle can be classified as (1) equilateral (2) isosceles (3) scalene (4) right 4. The angle measures of a triangle are 72°, 41°, and 67°. The triangle can be classified as (1) obtuse (2) right (3) isosceles (4) acute 5. A pair of adjacent angles in the accompanying figure are (1) ∠ABD and ∠CBD (2) ∠ABC and ∠CBA (3) ∠CBD and ∠ABC (4) ∠ABD and ∠ABC 6. Which is not a valid way to name the angle? (1) ∠STR (2) ∠RST (3) ∠TSR (4) ∠1 7. Which of the following can be used to describe the figure? (1) ∠Q (2) ∠FCQ and ∠GDC (3) intersects at Q (4) intersects at Q 8. Which of the following represents the three angles in ∆FLY? (1) ∠FLY, ∠LYF, ∠YLF (2) ∠FLY, ∠LYF, ∠YFL (3) ∠FLY, ∠LYF, ∠LYF (4) ∠LFY, ∠YFL, ∠YLF 9. Which of the following has a length? (1) a ray (2) a line (3) a segment (4) an angle 10. and are always (1) parallel segments (2) perpendicular segments (3) segments with reciprocal lengths (4) the same segment 11. Which of the following has an infinite length and width? (1) a point (2) a line (3) a segment (4) a plane B. Free Response—show all work or explain your answer completely 12. Sketch adjacent angles ∠DEF and ∠FEG. 13. Name all segments shown in the corresponding figure: 14. Name all angles shown in the corresponding figure. 1.2 BASIC RELATIONSHIPS AMONG POINTS, LINES, AND PLANES K I Geometric building blocks can be arranged in a number of ways relative to one another. These arrangements include parallel and perpendicular for lines and planes, and collinear for points. The relationships may be definitions, postulates, or theorems. A definition simply assigns a meaning to a word. A postulate is a statement that is accepted to be true but is not proven. A theorem is a true statement that can be proven. POSTULATES AND THEOREMS A definition assigns a meaning to a word using previously defined words. For example, “A triangle is a polygon with three sides.” Definitions provide only the minimum amount of information needed to define the word unambiguously. Properties that can be proven using the definition are not part of the definition. For example, in the definition of a triangle, we would not mention the fact that the angles in a triangle sum to 180°. That is a theorem that can be proven. A postulate or an axiom is a statement that is accepted to be true but cannot be proven. When proving a theorem, we cannot rely entirely on previously proven theorems because we need to start somewhere. Postulates are that starting point. Some of the postulates may seem obvious, so obvious in fact that the best one could do is to restate the postulate in diﬀerent words. For example, “Exactly one straight line may be drawn through two points” is a postulate. It is obviously true but cannot be proven using more fundamental postulates. A theorem is a statement that can be proven true using a logical argument based on facts and statements that are accepted to be true. If points, lines, and planes are the building blocks of geometry, then theorems are the cement that binds them together. Theorems o en express the relationships among the geometric figures and their measures that are the heart of geometry. An example of a theorem is “the diagonals of a square are perpendicular.” When proving a theorem, we may call upon previously proven theorems, postulates, and definitions. CONGRUENT The term congruent is similar to the term equal. However, congruent applies to geometric figures while equal applies to numbers. Figures that have the same size and shape are said to be congruent. The symbol for congruent is ≅. As o en happens in mathematics, there are diﬀerent approaches to determining if two figures are congruent. Since congruent figures have the same size and shape, we can compare lengths and angle measures. Two segments are congruent if their lengths are equal. Two angles are congruent if their angle measures are equal. Polygons are congruent if all pairs of corresponding angles and sides have the same measure. Circles are congruent if their radii are congruent. Alternatively, congruence can be established through transformations. Two figures are congruent if a set of rigid motion transformations map one figure onto the other. The transformation point of view is one that is emphasized in the Common Core and is discussed in detail in Section 5. Keep in mind the diﬀerence in notation between congruent and equality. If two segments, and , are congruent, we state that fact with. Since the segments are congruent, we know their lengths are equal, which we state with CD = EF. Note the diﬀerence in symbol, ≅ versus =. In addition, we use the overbar when referring to the segment and just the endpoints when referring to its length. Congruence of segments and angles can be specified in a sketch using tick marks for segments and arcs for angles. Sides with the same number of tick marks are congruent to one another, and angles with the same number of arcs are congruent to one another. Figure 1.10 shows a parallelogram with two pairs of congruent sides and two pairs of congruent angles. The pair of long sides each have one tick mark and are congruent, while the pair of short sides each have two tick marks and are congruent. The same is true for the two pairs of angles but using arcs. Figure 1.11 shows the congruent markings for a square. All four sides are congruent, so each side has one tick mark. The four angles are congruent, but they are also right angles, so the right angle marking can be used in place of the arcs. Figure 1.10 Congruent markings in a parallelogram Figure 1.11 Congruent and right angle markings in a square COLLINEAR AND COPLANAR A set of points that all lie on the same line is described as collinear. Figure 1.12a illustrates collinear points L, M, N, O. Points that are not collinear are described as noncollinear. Points R, S, and T in Figure 1.12b are noncollinear. Any two given points will always be collinear since a straight line can always be drawn through two points. This is a consequence of our first postulate. Figure 1.12 Collinear and noncollinear points Postulate 1 There is one, and only one, line that contains two given points. Extending to three dimensions, a set of points that all lie on the same plane is described as coplanar. Figure 1.13 illustrates coplanar points L, M, N, O. Points that do not lie on the same plane are noncoplanar. Any three given points will always be coplanar. Figure 1.13 Coplanar points L, M, N, O Postulate 2 There is one, and only one, plane that contains three given points. In addition to points, lines may also be coplanar. Coplanar lines are lines that are completely contained within the same plane. Remember, both the plane and the lines continue forever in their respective dimensions. INTERSECTING, PARALLEL, PERPENDICULAR, AND SKEW Coincide simply means to lie on top of one another. Two lines or planes that coincide are essentially the same. Intersecting means to cross one another. Intersecting lines always cross at a single point, called the point of intersection. Figure 1.14 shows lines r and s intersecting at point M. The intersection of two planes is always a single line, called the line of intersection. Figure 1.15 shows planes ABC and ABD intersecting at. Figure 1.14 Intersecting lines Figure 1.15 Intersecting planes Postulate 3 The intersection of two lines is a point. Postulate 4 The intersection of two planes is a line. Postulate 5 Intersecting lines are always coplanar. PERPENDICULAR Perpendicular is a special case of intersecting, where the lines or planes intersect at right angles. The symbol for perpendicular is ⊥. In Figure 1.16, line r ⊥ line s. In Figure 1.17, plane R ⊥ plane S. The small square at the right angle in Figure 1.16 is a symbol for a right angle. Segments and rays are perpendicular if the lines that contain them are perpendicular. Note that our definition of perpendicular involves right angles, not a 90° measure. Perpendicular lines lead us to right angles, and the right angles lead us to the 90° measure. Figure 1.16 Perpendicular lines Figure 1.17 Perpendicular planes PARALLEL Parallel lines are lines that never intersect and are coplanar. You can recognize parallel lines by the way they run in the same directions like a pair of train tracks. We use the symbol || for parallel. In Figure 1.18, line r || line s. Segments and rays are parallel if the lines that contain them are parallel. The “and are coplanar” part of the definition is important because it distinguishes parallel from skew. Planes can be parallel as well, as shown in Figure 1.19. Parallel planes never intersect. Figure 1.18 Line r || line s Figure 1.19 Plane R || Plane S SKEW LINES Skew lines are lines that are not coplanar. Like parallel lines, skew lines will never intersect. However, unlike parallel lines, skew lines run in diﬀerent directions. In Figure 1.20, line and line are skew. We do not have a special symbol for skew. Figure 1.20 Skew lines and M F Even though and are not connected with arrows in Figure 1.20, a line still exists that passes through each of the two pairs of points. Any two points can be used to specify a line. The same goes for planes. Plane ACGE slices diagonally through the prism even though we do not see the points connected in the manner seen in plane EFG. Any three points can be used to specify a plane. If you look at any pair of lines, one and only one of the following must be true. They can coincide, intersect, be parallel, or be skew. Any pair of planes will coincide, be parallel, or intersect. We do not use the word skew to describe planes. Examples For examples 1–5, use the figure of the cube below. 1. Identify 3 segments parallel to. 2. Identify 4 segments perpendicular to. 3. Identify 4 segments skew to. 4. Identify 1 plane parallel to plane EFG. 5. Identify 4 planes perpendicular to plane EFG. Solutions to examples 1–5: 1. , , and are parallel to. 2. , , , and are perpendicular to. 3. , , , and are skew to. 4. Plane ABC is parallel to plane EFG. 5. Planes EAB, FBC, GCD, and HDA are perpendicular to plane EFG. Check Your Understanding of Section 1.2 A. Multiple-Choice 1. Lines that are coplanar but do not intersect can be described as (1) perpendicular (2) parallel (3) skew (4) congruent 2. The intersection of two planes is (1) 1 point (2) 1 line (3) 2 points (4) 2 planes 3. Line r intersects parallel planes U and V. The intersection can be described as (1) 2 parallel lines (2) 1 line (3) 2 intersecting lines (4) 2 points 4. Points A, B, and C are not collinear. How many planes contain all three points? (1) one (2) two (3) three (4) an infinite number 5. In the figure of a rectangular prism, which of the following is true? (1) Points E, H, D, and A are coplanar and collinear. (2) is skew to , and. (3) , and. (4) , and skew to. 6. Which parts of the accompanying figure are congruent? (1) , ∠I ≅ ∠G, and ∠H ≅ ∠F (2) , , and ∠I ≅ ∠G (3) , , and ∠I ≅ ∠G (4) and ∠H ≅ ∠F 7. and intersect at point L. Which of the following is not true? (1) Points J, K, and M are collinear. (2) and are coplanar. (3) Points J, K, and L are collinear. (4) Points J, K, L, and M are coplanar. 8. Given points F, G, H, and I with no three of the points collinear, what is the maximum number of distinct lines that can be defined using points F, G, H, and I? (1) 4 (2) 5 (3) 6 (4) 8 9. Lines r and s intersect at point A. Line t intersects lines r and s and points B and C, respectively. Which of the following is true? (1) Lines r, s, and t must all be perpendicular. (2) Line t must be skew to lines r and s. (3) Points A, B, and C must be collinear. (4) Lines r, s, and t must all be coplanar. 10. If ∠J ≅ ∠L, which must be true? (1) m∠J = m∠L (2) ∠J ⊥ ∠L (3) ∠J || ∠L (4) m∠J + m∠L = 180° B. Free Response—show all work or explain your answer completely 11. In the triangular prism, a. name a segment skew to b. name two planes containing c. name a pair of parallel planes 12. Points M, N, and P are contained in both planes S and T. Juan states that the three points must be collinear, but his friend Carla disagrees and says they do not have to be. Who is correct? Explain your reasoning. 13. A stool has three legs, but one of the legs is shorter than the other two. When the stool is placed on a flat floor, will all three legs touch the floor? Explain why or why not. 1.3 BRIEF REVIEW OF ALGEBRA SKILLS K I Certain algebra skills show up frequently in our study of geometry. They are tools used to complete the evaluation of geometric relationships. Procedures for operations with radicals, solving linear equations, multiplying polynomials, solving quadratic equations, and solving proportions as well as word problem strategies are briefly reviewed. OPERATIONS WITH RADICALS The square root of a number is the number that when multiplied by itself results in the original number. It is represented with the square root, or radical, symbol. The number under the radical symbol is the radicand. An example of a radical expression is. The 3 is the coeﬀicient, and the 5 is the radicand. Adding and Subtracting Radicals: Add or subtract the coeﬀicient if the radicands are the same, otherwise the radicals cannot be combined. For example, only the terms can be combined in the following equation. Multiplying and Dividing Radicals: Multiply or divide the coeﬀicients and then multiply or divide the radicands. Simplifying Radicals—A radical is said to be in simplest form when the following 3 conditions are met. 1) No perfect square factors appear in the radicand. 2) No fractions appear in the radicand. 3) No radicals appear in the denominator. Remove perfect square factors from the radicand by factoring the radicand using the largest perfect square factor. Then take the square root of the perfect square factor. In the radical expression below, 12 is factored into 4 · 3. Then is simplified to 2. Fractions in the radicand can be rewritten as the quotient of two radicals as shown below. Remove radicals in the denominator by multiplying the numerator and denominator by the radical in the denominator. This is called “rationalizing the denominator.” M F Taking the square root and squaring are inverse operations. Taking the square root of a number and then squaring it results in the original number, as in. Example 1 Express in simplest radical form Solution: Example 2 Express in simplest radical form Solution: M F Expressing radicals in simplest radical form make it easier to compare radical expressions. In geometry, we o en want to determine if two measures are equal or satisfy a particular inequality. Once all the radicals have been completely simplified, comparing radicals is just a matter of comparing the coeﬀicients. Radicals will frequently show up when working with solving quadratic equations, when using the Pythagorean theorem, or with the distance formula. SOLVING LINEAR EQUATIONS Linear equations are equations that involve the variable raised to the first power only. They can be solved using the following steps. 1) Apply the distributive property to terms with parentheses. 2) Eliminate fractions by multiplying both sides by the denominator of any fraction, or the greatest common denominator if there are several fractions. 3) Combine like terms on each side of the equal sign. 4) Isolate the variable by undoing additions/subtractions and then multiplications. Example 1 Solve 8x + 6 = 2x + 4(x + 5) Solution: Example 2 Solve Solution: MULTIPLYING POLYNOMIALS When multiplying monomials, multiply the coeﬀicients and multiply the variables. When multiplying powers of the same variable, use the rule “keep the base and add the powers.” Example 1 Multiply Solution: When multiplying binomials, use the double distributive property by applying the vertical method, box method, FOIL (first-outer-inner-last), or any other technique you may have learned. Example 2 Multiply (4x + 2) (5x + 6) Solution: Use FOIL. Example 3 Multiply (3x − 7)(x + 2) Solution: Use the vertical method. FACTORING AND SOLVING QUADRATIC EQUATIONS Quadratic equations have second-order, or , terms as the highest power of x. Solving quadratic equations requires factoring. The procedure is as follows: Get all terms on one side. Factor. Apply the zero product rule. Solve for x. M F The zero product rule states that if a product of factors equals zero, then each factor may be individually set equal to zero and solved to find a solution to the equation. SOME FACTORING METHODS Greatest Common Factor: If a common factor exists among all terms, divide all terms by that factor. Put the new terms inside parentheses, and move the divided factor outside the parentheses. If a factor is still not linear, use another method on that factor. Example 1 Solve Solution: Grouping with a = 1: For the quadratic equation , find numbers and such that and. The equation factors to. From here, apply the zero product theorem. Example 2 Solve Solution: Diﬀerence of Perfect Squares: Quadratics in the form factor into Once factored, set each factor equal to zero and solve for x. Example 3 Solve Solution: Completing the Square: Completing the square can be used on any trinomial with the form. Rewrite the equation as Add the quantity to both sides of the equation. Factor the le side, which will be a perfect square. Take the square root of both sides, and solve for x. Remember, there will be a positive and negative root when taking the square root. So there will be two solutions. Example 4 Solve Solution: Quadratic Formula: The solution to any quadratic equation of the form can be found using the quadratic formula. Example 5 Solve Solution: M F Some geometric relationships result in a quadratic equation that must be solved in order to find the measure of an angle or segment. The quadratic will give two solutions, and both must be checked for consistency with the problem. Lengths or angle measure in this course will always be positive. If either solution results in a negative length or angle, that solution is thrown out. If both solutions lead to an acceptable answer, the problem has two solutions. Two solutions o en correspond to a situation where two diﬀerent geometric configurations could lead to the relationship modeled in the equation. SOLVING PROPORTIONS A proportion is an equation involving two ratios. They can be solved using the fact that the cross products must be equal. Example 1 Solve Solution: Example 2 Solve Solution: WORD PROBLEM STRATEGIES Word problems in geometry may involve phrases that describe a relationship between two figures or measures. Some common phrases and their algebraic translations are shown below. Phrase Algebra x is two more than y x=y+2 x is two greater than y x is two less than y x=y−2 x is twice y x = 2y x is double y x is half y Three quantities are in a 1 : 2 : 3 Represent the quantities as x, 2x, ratio and 3x The following are some good general strategies for solving word problems: 1) Make a sketch and label it. 2) Underline or highlight key words and definitions, such as bisector, midpoint, and so on. 3) Underline phrases to be translated into mathematical expressions. 4) Identify what the question is asking—the value of a variable, the measure of an angle or segment, an explanation or justification, and so on. Example 1 Write an expression that represents “12 less than double a number.” Solution: Let the number equal. Example 2 Three integers are in a 4 : 7 : 9 ratio. If their sum equals 60, what are the numbers? Solution: Let the integers equal 4x, 7x, and 9x. Check Your Understanding of Section 1.3 A. Multiple-Choice 1. is equal to (1) (2) (3) (4) 2. (1) 110 (2) 330 (3) 440 (4) 660 3. is equivalent to (1) (2) (3) (4) 4. The solution to is (1) x=0 (2) x=1 (3) x=2 (4) x=3 5. The solution to is (1) x = 3 (2) x = 6 (3) x = 9 (4) x = 12 6. The solution to the equation x2 − 12x + 20 = 0 is (1) x = −2, x = 10 (2) x = 2, x = 10 (3) x = −4, x = −5 (4) x = −20, x = 12 7. When factored, x2 − 36 is equal to (1) (x2 + 6)(6 − 12) (2) (x − 6)2 (3) (x + 6)(x − 6) (4) (x + 6)2 8. The length of a segment is given by the solution to x2 + 5x − 50 = 0. What are the possible lengths? (1) 5 only (2) 5 or 10 (3) 5 or −10 (4) 10 only 9. (1) (2) (3) (4) 8 10. Which expression represents “6 less than twice the measure of ∠1”? (1) 2 · m∠1 − 6 (2) 2 − 6 · m∠1 (3) 6 − 2 · m∠1 (4) 6 · m∠1 − 2 B. Free Response—show all work or explain your answer completely. All answers involving radicals should be in simplest radical form 11. Solve 3x2 − 27 = 0 by any method. 12. Solve x2 + 6x − 8 = 0 by completing the square. 13. Is less than, greater than, or equal to ? Justify your answer. 14. If AB is 3 greater than four times CD and the sum of the lengths is 33, find each length. 15. The measures of ∠A and ∠B sum to 180°, and m∠A is 9° greater than one-half m∠B. Find the measure of each angle. 16. The sides of a triangle are in a 3 : 5 : 6 ratio. If the perimeter has a length of 56, what is the length of the shortest side? 17. The measures of two angles sum to 90°, and they are in a ratio of 2 : 3. Find the measure of each angle. 18. Solve for x: 19. Solve for x: Chapter ANGLE AND TwoSEGMENT RELATIONSHIPS In this section, we explore some of the basic relationships involving the measures of angles and segments. Algebraic modeling and equation solving will be applied to find the measure of a missing angle or the measure of a segment. These relationships represent the building blocks that will be used to explore more complex problems, theorems, and proofs. 2.1 BASIC ANGLE RELATIONSHIPS K I Basic theorems and definitions used to solve problems involving angles include: The sum of the measures of adjacent angles around a point equals 360°. Supplementary angles have measures that sum to 180°. Complementary angles have measures that sum to 90°. Vertical angles are the congruent opposite formed by intersecting lines and are congruent. Angle bisectors divide angles into two congruent angles. The measure of a whole equals the sum of the measures of its parts. SUM OF THE ANGLES ABOUT A POINT The measures of the adjacent angles about a point sum to 360°. In Figure 2.1, m∠1 + m∠2 + m∠3 + m∠4 = 360°. Figure 2.1 Sum of the angles about a point = 360° Example Find the measure of each angle in the accompanying figure. Solution: SUPPLEMENTARY ANGLES, COMPLEMENTARY ANGLES, AND LINEAR PAIRS Supplementary angles are angles whose measures sum to 180°. The angles may or may not be adjacent. Two adjacent angles that form a straight line are called a linear pair. They are supplementary. Figure 2.2 shows a linear pair. Multiple adjacent angles around a line also sum to 180°, as shown in Figure 2.3. Figure 2.2 Linear pair Figure 2.3 Adjacent angles around a line Complementary angles are angles whose measures sum to 90°. As with supplementary angles, complementary angles may or may not be adjacent. Complementary angles are illustrated in Figure 2.4. Figure 2.4 Complementary angles Example 1 intersects at B. If m∠ABD = (4x + 8)° and m∠CBD = (2x + 4)°, find the measure of each angle. Solution: The angles form a linear pair and are supplementary. Example 2 In the accompanying figure, m∠1 = (x + 16)° and m∠2 = (3x + 6)°. Find the measure of each angle. Solution: The angles are complementary. VERTICAL ANGLES When two lines intersect, four angles are formed. Each pair of opposite angles are congruent and are called vertical angles. In Figure 2.5, ∠1 and ∠3 are vertical angles and are therefore congruent. Also, ∠2 and ∠4 are vertical angles and are therefore congruent. Vertical angles show up frequently in geometry. So always be on the lookout for them. Once you know the measure of any one of the four vertical angles formed by two intersecting lines, you can easily calculate the measures of the other three using the supplementary angle relationship for a linear pair. Figure 2.5 Vertical angles Example In Figure 2.5, m∠1 = 150°. Find the measure of each of the other angles. Solution: Vertical Angles Linear Pair m∠3 = m∠1 m∠4 + m∠1 = 180° m∠3 = 150° m∠4 + 150° = 180° Vertical Angles m∠4 = 30° m∠2 = m∠4 m∠2 = 30° Check Your Understanding of Section 2.1 A. Multiple-Choice 1. Two complementary angles measure (12x − 18)° and (5x + 23)°. What is the measure of the smaller angle? (1) 5° (2) 42° (3) 48° (4) 90° 2. Two supplementary angles measure (7x + 11)° and (14x + 1)°. What is the measure of the smaller angle? (1) 40.5° (2) 67° (3) 108° (4) 123° 3. and intersect at O. If m∠DOC = (8x − 30)° and m∠BOA = (6x + 12)°, what is m∠DOA? (1) 138° (2) 51° (3) 42° (4) 21° 4. What is the value of x in the accompanying figure? (1) 97° (2) 98° (3) 117° (4) 136° 5. Find the measure of ∠FIM. (1) 58° (2) 98° (3) 102° (4) 112° 6. Two lines intersect such that a pair of vertical angles have measures of (5x − 57)° and (3x + 21)°. Find the value of x. (1) 39 (2) 27 (3) 12 (4) 6.5 7. and m∠CFE = 42°. Find m∠DFA. (1) 132° (2) 138° (3) 142° (4) 144° 8. Lines and intersect at P. Which of the following is not necessarily true? (1) ∠APC and ∠CPB are supplementary. (2) ∠BPC ≅ ∠APD (3) ∠APC ≅ ∠BPD (4) ∠APC ≅ ∠BPC B. Free Response—show all work or explain your answer completely 9. In the accompanying figure, , intersects at point O, and m∠JOH = 31°. Find the measures of ∠KOG and ∠IOH. 10. Two supplementary angles are congruent. What is the measure of each? 11. The measure of ∠A is 15° greater than twice the measure of ∠B. If ∠A and ∠B are complementary, what are the measures of ∠A and ∠B? 12. intersects at R. If the measure of ∠QRU increases by 15°, what is the change in the measure of ∠SRT and ∠SRU? 2.2 BISECTORS, MIDPOINT, AND THE ADDITION POSTULATE K I Whenever a segment or an angle is divided into two or more smaller parts, the sum of the smaller parts equals the original part. This is called the segment or angle addition postulate. When the part is divided exactly in half, we say it is bisected and the two parts are congruent to each other. An angle bisector is a line, segment, or ray that divides an angle into two congruent parts. A segment bisector intersects a segment at its midpoint, which divides it into two congruent parts. SEGMENT AND ANGLE ADDITION Any segment can be divided by locating a point between the two endpoints, creating two new segments. The segment addition postulate states a segment’s length equals the sum of the two new segments formed by any point between the endpoints of the original segment. As shown in Figure 2.6, point C is located on between A and B. Therefore, AC + CB = AB. Figure 2.6 Segment addition Example 1 Point S is located between points R and T on. If RS = (2x + 1), ST = (3x + 4), and RT = 35, find the lengths of RS and ST. Solution: You should always sketch the figure and write the segment addition postulate in terms of segment names. This will give you the opportunity to check the postulate against the figure. If something does not make sense, you can make a correction before going further. Once you have substituted in algebraic expressions, it is much more diﬀicult to check whether the equation makes sense from a geometry point of view. The angle addition postulate is similar to the segment addition postulate. The measure of an angle is equal to the sum of the two adjacent angles formed by an interior ray with its endpoint at the vertex of the original angle. Example 2 m∠ABC = (6x − 7)°, m∠ABD = (3x + 3), and m∠CBD = (2x + 12). Find m∠ABC. Solution: Besides addition, we also have postulates for segment subtraction and angle subtraction. These are used in a manner similar to segment and angle addition. Example 3 Write an expression for the length of EB in terms of lengths AB and AE. Solution: ANGLE BISECTORS A line, segment, or ray that divides an angle into two congruent angles is called an angle bisector. Figure 2.7 illustrates an angle bisector. Ray bisects ∠ACB. The two congruent angles formed are ∠ACD and ∠BCD. Figure 2.7 Angle bisector Example bisects ∠ABD. If m∠ABD = (8x − 12)° and m∠ABC = (3x + 4)°, find the value of x. Solution: MIDPOINTS AND SEGMENT BISECTORS In segments, the point that divides a segment into two congruent segments is called a midpoint. Any line, segment, or ray that intersects a segment at its midpoint is called a segment bisector, or simply bisector. Figure 2.8 illustrates a segment bisector. Line m bisects at its midpoint S, and. Note that lines can bisect segments. However, lines cannot be bisected themselves because lines do not have a finite length. Figure 2.8 Segment bisector m M F In geometry, the definition of a segment bisector, or simply bisector, is “a segment, line, or ray that passes through the midpoint of a segment.” The definition says nothing about ending up with two congruent segments. It is the definition of a midpoint that tells us we end up with two congruent segments. A midpoint on a segment will always give two relationships, the pair of congruent parts and the segment addition postulate, as seen in Figure 2.9. One or both relationships may be needed to solve a problem. Figure 2.9 Segment relationships with a midpoint Example 1 J is the midpoint of. GJ = (7x + 1) and JH = 3x + 9. Find GH. Solution: Example 2 bisects at point W. If PW = 2x + 7 and PQ = 3x + 24, find the length of PW. Solution: A special case of a bisector is the perpendicular bisector. This is a segment, line, or ray that passes through a segment’s midpoint at a right angle. Figure 2.10 shows perpendicular bisector passing through midpoint E at a right angle. Figure 2.10 Perpendicular bisector Check Your Understanding of Section 2.2 A. Multiple-Choice 1. Point R is the midpoint of , and point S is the midpoint of. Which of the following is true? (1) UR = RS (2) (3) UR + RS = SV (4) UR = 2(UV) 2. In segment , CD = 2x + 7, DF = 3x + 6, and CF = 9x − 11. What is the length of CF? (1) 6 (2) 9 (3) 24 (4) 43 3. Point X is the midpoint of segment. If BX = 3x − 4 and XM = x + 12, what is the length of BM? (1) 5 (2) 8 (3) 40 (4) 80 4. m∠ABD = (2x + 8)°, m∠DBC = (3x + 12)°, and m∠ABC = (8x − 1)°. What is the measure of ∠ABC? (1) 55° (2) 57° (3) 61° (4) 63° 5. ∠FGH is bisected by. If m∠FGK = (4x + 5)° and m∠FGH = (14x − 20)°, what is the value of x? (1) 1 (2) 2.5 (3) 3.5 (4) 5 6. In segment , S is between R and U, and T is between S and U. Which relationship must be true? (1) RS = ST (2) RS + ST = TU (3) RU − TU = RT (4) ST − TU = RS 7. ∠SAR is bisected by. Which relationship must be true? (1) m∠SAT = m∠SAR (2) m∠TAR = m∠RAS (3) (4) m∠SAR = 2m∠TAR 8. Perpendicular bisector intersects at A. Which of the following is not necessarily true? (1) (2) (3) ∠UAX ≅ ∠XAV (4) ∠WAV ≅ ∠UAX 9. Given , m∠VXY = 144° and m∠ZXW = 138°. What is the measure of ∠ZXV? (1) 36° (2) 42° (3) 90° (4) 102° B. Free Response—show all work or explain your answer completely 10. In △JKL, is a median. If JM = (8x − 21) and ML = (2x + 33), find JL. 11. intersects at P. If AP = (7x − 14), BP = (3x + 6), and m∠BPD = (8y − 6)°, for what values of x and y is the perpendicular bisector of ? 12. In the accompanying figure, m∠TRA = 62°, m∠ARP = (5x + 1)°, and m∠PRT = (10x + 3)°. Determine if is an angle bisector. Explain your reasoning. 2.3 ANGLES IN POLYGONS K I There are relationships between the sides of a polygon with n sides and the measures of its interior and exterior angles. The sum of the exterior angles of a polygon always equals 360°. The sum of the interior angles of a polygon equals 180°(n − 2), where n equals the number of sides. A central angle of a regular polygon equals 360°/n, where n equals the number of sides. INTERIOR ANGLES OF POLYGONS An interior angle of a polygon is the interior angle formed by the intersection of two consecutive sides. Figure 2.11 shows the 4 interior angles of a quadrilateral. Figure 2.11 Interior angles in a quadrilateral A regular polygon is a polygon in which all the interior angles and all the sides are congruent. The sum of the interior angles of any polygon, and measure of a single interior angle of a regular polygon can be found using the following theorem. I A T P The sum of the measures of the interior angles of a polygon with n sides is given by 180°(n − 2). The measure of one interior angle of a regular polygon with n sides is given by. Example 1 What is the measure of each interior angle of a regular hexagon? Solution: For a hexagon, n = 6. So each interior angle in a regular hexagon measures Example 2 The sum of the measures of the interior angles of a polygon equals 540°. How many sides does the polygon have? Solution: Example 3 In regular hexagon ABCDEF,. Find m∠CBD. Solution: M T When faced with a complex problem, break it up into smaller pieces that you recognize. Then begin working your way toward your goal. EXTERIOR ANGLES IN POLYGONS An exterior angle of a polygon is formed by extending one side of the polygon. We consider only one extended side at each vertex, so a polygon with n sides has n exterior angles. Figure 2.12 illustrates the 5 exterior angles of a pentagon. Which side is extended does not matter because extending the second side at any vertex would result in a pair of vertical angles with equal measures. Figure 2.12 Exterior angles in a pentagon For regular polygons, the measure of one exterior angle equals , where n equals the number of sides. E A T P The sum of the measures of the exterior angles of a polygon with n sides equals 360°. The measure of one exterior angle of a regular polygon with n sides equals. Example What is the measure of one exterior angle of a regular decagon? Solution: Each exterior angle measures and n = 10. CENTRAL ANGLES A central angle is an angle with its vertex at the center of the polygon and rays through consecutive vertices, as shown in Figure 2.13. Central angles of a polygon are found in the same way as exterior angles. Figure 2.13 Central angle ∠APB C A F The sum of the central angles in a polygon is 360°. The measure of each central angle in a regular polygon equals , where n is the number of sides. M F If a regular polygon has an even number of sides, the center is the point of concurrency of segments with opposite vertices as their endpoints. If the polygon has an odd number of sides, the center is the point of concurrency of segments with a vertex at one endpoint and the midpoint of the opposite side as the other endpoint. Example Find the measures of angles x and y in a regular pentagon with center P, as shown in the accompanying figure. Solution: In a pentagon, n = 5. So each central angle measures and y = 72°. Each interior angle of the pentagon measures. m∠x is the measure of an interior angle, or 54°. Check Your Understanding of Section 2.3 A. Multiple-Choice 1. What is the sum of the interior angles of a pentagon? (1) 108° (2) 360° (3) 540° (4) 720° 2. How many sides does a polygon have whose interior angles sum to 1,800°? (1) 6 (2) 8 (3) 10 (4) 12 3. What is the measure of one central angle of an equilateral triangle? (1) 60° (2) 90° (3) 120° (4) 150° 4. What is the measure of one central angle of a regular octagon? (1) 45° (2) 60° (3) 72° (4) 144° 5. What is the measure of one exterior angle of a regular hexagon? (1) 60° (2) 120° (3) 135° (4) 144° 6. The difference between one interior and one exterior angle of a regular polygon is 90°. How many sides does the polygon have? (1) 4 (2) 5 (3) 6 (4) 8 7. In a regular polygon, adjacent interior and exterior angles are congruent. What is the name of the polygon? (1) triangle (2) rectangle (3) square (4) hexagon 8. A regular octagon and a regular hexagon have equal side lengths and share a side. What is the measure of the angle indicated by x in the figure? (1) 10° (2) 15° (3) 20° (4) 33° B. Free Response—show all work or explain your answer completely 9. Point O is the center of the accompanying figure. Find the measure of angle x. 10. Find the value of x in the accompanying figure. 11. In the accompanying hexagon, two pairs of sides are extended. Using the angle measures provided, find the value of x. 12. A regular pentagon and a regular octagon have equal side lengths and share a side as shown. What is the value of x? 2.4 PARALLEL LINES K I When two or more parallel lines are intersected by a transversal, any pair of angles formed are either congruent or supplementary. We can determine if two lines are parallel by looking at the measures of these pairs. Two lines are parallel if and only if the following are true: Alternate interior angles are congruent. Same side interior angles are supplementary. Corresponding angles are congruent. NAMED ANGLE PAIRS Whenever two lines are intersected by another line, called a transversal, the eight angles shown in Figure 2.14 will be formed. Some of the angle pairs have names to make referencing them easier. Notice that the transversal divides the figure into two regions, on alternate sides of the transversal. In Figure 2.14, angles 1, 4, 5, and 8 are on one side. Angles 2, 3, 6, and 7 are on the other side. Next, the two lines intersected by the transversal form an interior region and an exterior region. Angles 3, 4, 5, and 6 are interior angles, while angles 1, 2, 7, and 8 are exterior angles. Figure 2.14 The 8 angles formed when two lines are intersected by a transversal Using these definitions, the named pairs are: Alternate interior angles are ∠3 and ∠5 and also ∠4 and ∠6. Same side interior angles are ∠4 and ∠5 and also ∠3 and ∠6. Corresponding angles are ∠4 and ∠8, are ∠1 and ∠5, are ∠2 and ∠6, and also ∠3 and ∠7. Alternate exterior angles are ∠1 and ∠7 and also ∠2 and ∠8. RECOGNIZING ANGLE PAIRS Alternate interior angles make a “Z” pattern that can help in recognizing them. The alternate interior angles are in the corners of the “Z,” as shown in Figure 2.15. Same side interior angles make an “F” pattern. Those angles are in the corners of the “F” as shown in Figure 2.16. Figure 2.15 Alternate interior angles in the corners of the “Z” Figure 2.16 Same side interior angles in the corners of the “F” THEOREMS INVOLVING PARALLEL LINES AND THE NAMED ANGLE PAIRS When parallel lines are intersected by a transversal, a set of relationships is formed among the angles. Corresponding Angle Postulate Two lines intersected by a transversal are parallel if and only if the corresponding angles are congruent. Alternate Interior Angle Theorem Two lines intersected by a transversal are parallel if and only if the alternate interior angles are congruent. Same Side Interior Angle Theorem Two lines intersected by a transversal are parallel if and only if the same side interior angles are supplementary. Alternate Exterior Angle Theorem Two lines intersected by a transversal are parallel if and only if the alternate exterior interior angles are congruent. All of the angle relationships involving parallel lines result in congruent pairs of angles or supplementary pairs of angles. Combining these three angle theorems with the linear pair and vertical angle postulates allows you to calculate the measure of all eight of the angles if the measure of just one angle is known. Example 1 Line m || line n and m∠1 = 118°. Find the measures of the other angles. Solution: m∠1 + m∠2 = 180° linear pair is supplementary 118° + m∠2 = 180° m∠2 = 62° m∠3 = m∠1 = 118° vertical angles are congruent m∠4 = m∠2 = 62° vertical angles are congruent m∠6 = m∠2 = 62° corresponding angles are congruent m∠7 = m∠3 = 118° corresponding angles are congruent m∠5 = m∠7 = 118° vertical angles are congruent m∠8 = m∠4 = 62° corresponding angles are congruent Example 2 Line m || line n. Find the value of x. Solution: The indicated angles are supplementary, same side interior angles. Example 3 Line m || line n. What is the value of x? Solution: alternate interior angles are congruent Example 4 , m∠A = 32°, and m∠B = 38°. Find m∠ACD and m∠BCE. Solution: Use the two pairs of alternate interior angles. USING AN AUXILIARY LINE Sometimes we need to construct an additional line, called an auxiliary line, to help solve a parallel line problem. Example 1 , m∠CDE = 38°, and m∠ABE = 32°. Find m∠BED. Solution: We need to construct an auxiliary line, , through point E and parallel to and. congruent alternate interior angles congruent alternate interior angles angle addition Example 2 Lines r and s are parallel. Find m∠1. Solution: There are two “jogs” in the transversal. So we need two auxiliary lines, both parallel to lines r and s. m∠5 = 38° alternate interior angles m∠4 = 64° − 38° = 26° angle subtraction m∠3 = m∠4 = 26° alternate interior angles m∠2 = 24° alternate interior angles m∠1 = m∠2 + m∠3 = 50° angle addition DETERMINING IF TWO LINES ARE PARALLEL The relationships between corresponding, alternate interior, and same side interior angles are true when read in either direction. For example, the corresponding angle postulate tells us two things: If two lines are parallel, then the corresponding angles are congruent. If the corresponding angles are congruent, then the lines are parallel. The second statement can be used to determine whether two lines are parallel by comparing the measures of any pairs of corresponding angles. The same can be done for alternate interior and same side interior angles. Example 1 Lines m and n are intersected by transversal v, m∠4 = 52°, and m∠5 = 128°. Determine if lines m and n are parallel. Solution: Same side interior angles ∠4 and ∠7 are supplementary, so lines m and n are parallel. Example 2 In quadrilateral ABCD, m∠A = 111°, m∠B = 69°, m∠C = 107°, and m∠D = 73°. How many pairs of parallel sides does ABCD have? Solution: because same side interior angles are supplementary. m∠A + m∠B = 69° + 111° = 180° and are not parallel because the same side interior angles are not supplementary. So quadrilateral ABCD has 1 pair of parallel sides. Example 3 For what value of x is ? Solution: ∠AEF and ∠CFH are corresponding angles. They must be congruent if. Check Your Understanding of Section 2.4 A. Multiple-Choice Use the following figure for problems 1–5. 1. Line r || line s, m∠2 = (7x + 18)°, and m∠8 = 13x. What is m∠2? (1) 3° (2) 13° (3) 39° (4) 141° 2. Line r || line s, m∠4 = (3x + 34)°, and m∠6 = (2x + 46)°. What is m∠6? (1) 12° (2) 20° (3) 70° (4) 86° 3. Line r || line s and m∠3 = 118°. What is m∠6? (1) 32° (2) 62° (3) 118° (4) 360° 4. Line r || line s, m∠2 = (4x − 27)° and m∠6 = (x + 12)°. What are the measures of ∠2 and of ∠6? (1) m∠2 = 39° and m∠6 = 141° (2) m∠2 = 51° and m∠6 = 129° (3) m∠2 = 25° and m∠6 = 155° (4) m∠2 = 25° and m∠6 = 25° 5. Line r || line s, m∠4 = (4x + 35)°, and m∠7 = (6x + 25)°. What is the measure of ∠4? (1) 5° (2) 12° (3) 55° (4) 83° 6. Jay St. and Clayton St. run parallel to each other. Henry St. is straight and intersects both Jay St. and Clayton St. If the obtuse angle formed by Jay St. and Henry St. measures (3x + 51)° and the measure of the acute angle formed by Clayton St. and Henry St. is (4x + 3)°, what is the measure of the acute angle formed by Jay St. and Henry St.? (1) 18° (2) 48° (3) 52° (4) 75° 7. intersects and. If m∠AEF = (2x + 5)° and m∠CFE = (2x + 35)°, for what value of x are and parallel? (1) 4.5 (2) 6.5 (3) 35 (4) 69 8. Line m || line n. Find the value of x. (1) 19° (2) 41° (3) 60° (4) 68° 9. Find the measure of ∠1. (1) 63° (2) 117° (3) 95° (4) 32° 10. Which of the following would be a justification for proving two lines are parallel? (1) A pair of corresponding angles are supplementary. (2) A pair of alternate interior angles are supplementary. (3) A pair of same side interior angles are supplementary. (4) A pair of same side interior angles are congruent. B. Free-Response—show all work or explain your answer completely 11. In the accompanying figure, , m∠CDA = (4x + 10)°, m∠ADB = (4x)°, m∠DBA = (3x − 6)°, and m∠DEB = (5x − 6)°. Is ? Justify your reasoning. 12. Lines s and t are parallel. Find m∠1. 13. Lines s and t are parallel. Find the measure of ∠1. 14. Lines s and t are parallel. Find the measure of angle 1. 15. A pair of parallel lines intersected by a transversal forms same side interior angles that are in a 5 : 1 ratio. What are the measures of two same side interior angles? 16. Are rays a and b parallel? Justify your reasoning. 2.5 ANGLES AND SIDES IN TRIANGLES K I A set of theorems relate angle measures in triangles. The sum of the measures of the interior angles of a triangle equals 180°. The measure of any exterior angle of a triangle equals the sum of the measures of the nonadjacent interior angles. If two sides of a triangle are congruent, then the angles opposite them are congruent. All interior angles of an equilateral triangle measure 60°. TRIANGLE ANGLE SUM THEOREM The angle sum theorem for triangles is simply the polygon interior angle theorem evaluated for a polygon with three sides. The sum of the interior angles of a triangle equals 180°. M F The triangle angle sum theorem can be used to help classify the type of triangle. The theorem is used to calculate the measures of each of the three angles in a triangle. Example 1 The measures of the three angles in a triangle are (3x + 12)°, (4x − 6)°, and (2x + 12)°. Find the measure of each angle, and completely classify the type of triangle. Solution: Set the sum of the angle measures equal to 180°. The three angle measures are: The triangle is isosceles because two angles are congruent, and acute because all angles measure less than 90°. Example 2 In △LOV, m∠LOE = (5x − 68)°, m∠VOE = (2x + 7)°, m∠V = 28°, and is an angle bisector. Find m∠L. Solution: By substituting x into the expressions for each angle, we find: angle addition angle sum theorem in △LOV TRIANGLE EXTERIOR ANGLE THEOREM The triangle exterior angle theorem relates the measure of an exterior angle of a triangle to the two nonadjacent interior angles. In Figure 2.17, m∠1 = m∠2 + m∠3. Figure 2.17 Exterior angle 1 and nonadjacent angles 2 and 3 This theorem is particularly useful when exterior angles are expressed in terms of a variable. Example 1 In △ABC, side is extended through point A to point D. If m∠CAD = 140° and m∠ACB = 95°, find m∠ABC. Is △ABC isosceles? Solution: From the exterior angle theorem, exterior angle theorem linear pair △ABC is not isosceles since no two angles in the triangle are congruent. Example 2 In △RST, m∠URT = (10x + 39)°, m∠RST = (7x + 4)°, and m∠STR = (6x + 8)°. Find m∠URT. Solution: Apply the triangle exterior angle theorem, ISOSCELES TRIANGLES An isosceles triangle has two congruent sides, and the angles across from those sides are always congruent. The congruent angles are called the base angles, and the congruent sides are called the legs. The side that diﬀers in length is called the base, and the angle that diﬀers in measure is called the vertex angle. These parts are shown in Figure 2.18. Figure 2.18 Isosceles triangle I T T If two sides in a triangle are congruent, then the angles opposite those sides are congruent. C I T T If two angles in a triangle are congruent, then the sides opposite those angles are congruent. If any one angle of an isosceles triangle is known, then the other angles can be determined. Example 1 In Figure 2.18, m∠C = 106°. Find m∠A and m∠B. Solution: Base angles ∠A and ∠B are congruent. Use the triangle angle sum theorem, Example 2 In Figure 2.18, m∠A = 42°. Find m∠C. Solution: Base angles ∠A and ∠B are congruent. Example 3 In the accompanying figure, and. If m∠1 = 68°, what is m∠2? Solution: triangle angle sum theorem Example 4 In the accompanying figure, , m∠ACD = (9x − 4)°, and m∠ABC = (5x − 10)°. Find m∠A. Solution: △ABC is isosceles with ∠B ≅ ∠A. Use the exterior angle theorem, EQUILATERAL TRIANGLES An equilateral triangle is one in which all three sides are congruent and all three angles measure 60°. Example △ABC is equilateral, and D lies on. m∠ADC = 102°. Find m∠BAD. Solution: angles in an equilateral triangle measure 60° angle sum theorem in △ADC angle addition PYTHAGOREAN THEOREM In a right triangle, the sum of the squares of the legs is equal to the square of the hypotenuse, as shown in Figure 2.19. The hypotenuse is always the longest leg of the triangle, which is opposite the 90° angle. Remember, the Pythagorean theorem applies only to right triangles. Figure 2.19 Pythagorean theorem In a right triangle with legs represented by lengths a and b and with hypotenuse c, a2 + b2 = c2. Example 1 In △RST, ∠S measures 90°, RS = 6, and ST = 3. Find RT. Solution: Since ∠S is the right angle, RT must be the hypotenuse. Example 2 In the accompanying figure, m∠E = 45°, EO = GO, and EG = 8. Find EO and GO. Solution: isosceles triangle theorem angle sum theorem △GEO is a right triangle, so Pythagorean theorem applies let EO = GO = x Example 3 The length of a rectangle is 3 times its width. If the length of a diagonal is 10, what is the area of the rectangle? Solution: The right triangle formed by the sides of the rectangle and a diagonal has legs equal to 3x and x and a hypotenuse equal to 10. Example 4 In △ABC, m∠A = 90°, AC = 6, AB = 2x, and BC = x + 6. Find the area of the triangle. Solution: Check Your Understanding of Section 2.5 A. Multiple-Choice 1. In the accompanying figure, m∠I = 32°, m∠H = 76°, and m∠G = 78°. Find m∠E. (1) 24° (2) 30° (3) 72° (4) 78° 2. In △MNO, side MN is extended to point P. m∠PNO = (6x + 41)°, m∠NMO = (2x + 20)°, and m∠MON = (x + 39)°. What is the measure of ∠MNO? (1) 32° (2) 45° (3) 77° (4) 103° 3. In △ABD with side , , m∠BAC = 38°, and m∠D = 61°. Find m∠DAC. (1) 20° (2) 23° (3) 29° (4) 71° 4. In △ABC, AC = BC. If m∠A is four times m∠C, find m∠B. (1) 20° (2) 60° (3) 80° (4) 144° 5. The perimeter of an equilateral triangle is (6x − 12) inches. If each side measures 24 inches, what is the value of x? (1) 6 (2) 14 (3) 32 (4) 72 6. The measures of the angles of a triangle are in a 2 : 5 : 8 ratio. The measure of the smallest angle is (1) 6° (2) 12° (3) 24° (4) 96° 7. The measures of the angles of a triangle are (3x − 4)°, (2x + 15)°, and (10x + 4)°. The triangle can be classified as (1) isosceles acute (2) scalene acute (3) isosceles obtuse (4) scalene obtuse 8. In △FIS, side IS is extended to point H. Find the measure of ∠F. (1) 11° (2) 18° (3) 19° (4) 30° 9. In △FGH, m∠F = 35°, m∠G = (x + 21)°, and m∠H = (3x − 20)°. Find the measure of ∠H. (1) 18° (2) 36° (3) 57° (4) 88° 10. Two legs of a right triangle measure 4 and 8. What is the length of the hypotenuse? (1) (2) (3) (4) B. Free-Response—show all work or explain your answer completely 11. , m∠DCA = 78°, and AC = BC to AC = BC. Find m∠ACB. 12. Is a triangle whose side lengths measure 5, 6, and 8 a right triangle? Justify your answer. 13. In the accompanying figure, m∠GJI = 132°, , and. Find m∠F. 14. In rectangle JUMP with diagonal JM, JU = 8 and JM = 12. Find the length of UM. Give your answer in simplest radical form. 15. To get home from school, Stanasha has to walk 5 blocks north and 12 blocks east. Her school provides free bus passes for any student who lives more than mile from the school, as measured by the straight-line distance. If 20 blocks equals 1 mile, does Stanasha qualify for a free bus pass? 16. The hypotenuse of a right triangle measures. If the length of one leg is 5, what is the length of the second leg? Give your answer in simplest radical form. 17. The area of an isosceles right triangle is 32. What is the length of a leg of the triangle? 18. The length of the diagonal of a square measures 10 cm2. What is the perimeter of the square? Express your answer in simplest radical form. 19. Corey is building a new rectangular deck with a length of 24 feet and a width of 10 feet. He marks the location of the four corner support posts and uses a tape measure to confirm the lengths and widths are correct. Explain how he can use the tape measure to check if the angles are 90°. 20. Point P is the center of regular pentagon ABCDE. Find m∠PAC. 21. In the accompanying figure, , , and △MNP is equilateral. Find m∠PNO. 22. , , m∠JAO = 126°, and m∠BOA = 18°. Find m∠JBO. Chapter CONSTRUCTIONS Three 3.1 BASIC CONSTRUCTIONS K I A construction uses a straightedge and a compass to create a precise geometric figure. The compass is used to construct arcs whose points are all equidistant from the center point. Intersecting arcs locate a point equidistant from the two centers. Two intersecting circles create points equidistant from the two centers. The fundamental loci described in Section 3.1 are used to create a variety of constructions from these two building blocks. Some of the basic constructions are the equilateral triangle, perpendicular line, and parallel line. DEFINITION OF CONSTRUCTION A construction is a geometric drawing in which only a straightedge and compass may be used. The straightedge is used only for connecting two points in a straight line. If your straightedge is a ruler, you cannot use the length markings on it to measure length. When constructing an angle of a particular measure, such as a 45° angle, you cannot use a protractor to measure the angle. The only measuring device allowed is the compass, and we will use it to measure both length and angle opening. THE COMPASS The compass is simply a tool for constructing circles. The point of the compass locates the center point P, and the distance the compass is opened defines the radius. As long as the opening of the compass does not change, every point it traces out is a fixed distance from the center, resulting in a circle. You should purchase a good-quality compass. A compass that is too loose or not rigid enough will be very frustrating to use. Tips for using a compass Adjust the height of the pencil so that the compass stands nearly vertical when the point and pencil are touching the paper. Experiment with diﬀerent techniques for holding the compass and making arcs. One technique worth trying is to hold the compass from only the point end and rotate the paper instead of the compass. Some compasses have a screw at the pivot that can be tightened. Tighten this screw if you find the compass opening and changing as you make an arc. All arcs shown in the following constructions are called the “marks of construction.” You must include them when doing a construction on an exam; they are required. These marks demonstrate that you followed the proper procedure with the compass. COPY A SEGMENT Given: and point C, construct congruent to. Procedure: 1) Place compass point on A and pencil on B; make a small arc. 2) With the same compass opening, place point on C and make an arc. 3) Use the straightedge to connect point C to any point D on the arc,. COPY AN ANGLE Given: Angle ∠ABC and , construct ∠DEF congruent to ∠ABC. Procedure: 1) Place compass point on B, and make an arc intersecting the angle at R and at S. 2) With the same compass opening, place point on E and make an arc intersecting at T. 3) Place compass point on R and pencil on S, and make a small arc. 4) With the same compass opening, place point on T and make an arc intersecting the previous one at F. 5) Use the straightedge to form. 6) ∠DEF is congruent to ∠ABC. EQUILATERAL TRIANGLE Given: Segment , construct an equilateral triangle with side length AB. Procedure: 1) Place compass point on A and pencil on B. Make a quarter circle. 2) Place compass point on B and pencil on A. Make a quarter circle that intersects the first at C. 3) Use a straightedge to form AC and BC. ΔABC is an equilateral triangle. ANGLE BISECTOR Given: Angle ∠ABC, construct angle bisector. Procedure: 1) Place compass point on B, and make an arc intersecting and at R and S, respectively. 2) Place compass point on R, and make an arc in the interior of the angle. 3) With the same compass opening, place compass point on S and make an arc that intersects the previous arc at point D. 4) Use the straightedge to form. is the angle bisector. PERPENDICULAR BISECTOR Given: Segment , construct the perpendicular bisector of. Procedure: 1) With the compass open more than half the length of , place the point at A and make a semicircle running above and below. 2) With the same compass opening, place the point at B and make a semicircle running above and below. Make sure the semicircle intersects the first semicircle at R and at S. 3) Use a straightedge to connect R and S. is the perpendicular bisector of. Check Your Understanding of Section 3.1 A. Multiple-Choice 1. Which of the following is not necessarily true in the construction shown in the accompanying figure? (1) m∠RAD = m∠CAD (2) m∠CAD + m∠RAD = m∠CAR (3) AC ≌ AD (4) AS ≌ AT 2. Which of the following must be true in the accompanying construction? (1) m∠C is greater than m∠A. (2) There is not enough information in the figure to determine the measure of ∠ABC. (3) Point C is equidistant from points A and B. (4) AB is greater than BC and AC. 3. Frankie is constructing a copy of ∠SAD. The accompanying figure shows his progress so far. What should his next step be? (1) Place the point of his compass at M, and make an arc intersecting at D. (2) Place the point of his compass at S, and make an arc intersecting at D. (3) Place the point of his compass at A, and make an arc intersecting at N. (4) Place the point of his compass at M, and make an arc intersecting at N. B. Free Response—show all work or explain your answer completely 4. Construct congruent to. 5. Construct a segment whose length is equal to 3AR. 6. Construct a segment whose length is equal to MA + TH. 7. Construct ∠FGH congruent to ∠CDE. 8. Construct an angle whose measure is twice that of ∠LMN. 9. Bisect ∠LMN above. 10. Construct a right angle. 11. Construct a 45° angle. 12. Construct an equilateral triangle with as one of its sides. 13. Construct a 30° angle. 14. Construct the midpoint of. 3.2 CONSTRUCTIONS THAT BUILD ON THE BASIC CONSTRUCTIONS K I The basic constructions of the previous section can be combined to produce a number of other constructions. PERPENDICULAR TO LINE FROM A POINT NOT ON THE LINE Given line and point P not on , construct a line perpendicular to passing through P. Procedure: 1. Place compass point at P, and make an arc intersecting at R and S. (Extend if necessary.) 2. Place compass point at R, and make an arc on the opposite side of the line as P. 3. With the same compass setting, place compass point at S, and make an arc intersecting the previous arc at Q. 4. Use the straightedge to connect P and Q. is perpendicular to and passes through P. PERPENDICULAR TO LINE FROM A POINT ON THE LINE Given line and point P on between A and B, construct a line perpendicular to and passing through P. Procedure: 1. Place compass point at P, and make an arc intersecting at R and at S. 2. Place compass point at R, and make an arc below. 3. With the same compass setting, place point at S and make an arc intersecting the previous arc at Q. 4. Use a straightedge to connect P and Q. is perpendicular to and passes through P. PARALLEL TO LINE FROM POINT OFF THE LINE Given: Segment and point P not on , construct a line parallel to and passing through P. Procedure: 1. Use the straightedge to construct a line passing through P and intersecting at R. 2. With compass point at R, make an arc intersecting and at S and T, respectively. 3. With the compass at the same setting and compass point at P, make an arc intersecting at U. 4. With compass point at T, make an arc intersecting at S. 5. With the compass at the same setting and compass point at U, make an arc intersecting at V. 6. Use a straightedge to connect P and V. is parallel to and passes through point P. INSCRIBE A REGULAR HEXAGON OR AN EQUILATERAL TRIANGLE IN A CIRCLE Procedure: 1. Construct circle P of radius PA. 2. Using the same radius as the circle, place compass point on A and make an arc intersecting the circle at B. 3. With the same compass setting, place compass point at B and make an arc intersecting the circle at C. Continue making arcs intersecting at D, E, and F. 4. Using the straightedge, connect each point on the circle to form regular hexagon ABCDEF. Connecting every other point will form equilateral triangle ACE. INSCRIBE A SQUARE IN A CIRCLE Procedure: 1. Construct a diameter through the center point T of a circle. 2. Construct the perpendicular bisector of the first diameter, which will also be a diameter. 3. The intersections of the diameters with the circle are the vertices of the square. 4. Use a straightedge to connect the vertices and form the sides of the square. Check Your Understanding of Section 3.2 A. Multiple-Choice 1. Jamie is constructing a line perpendicular to. Her progress so far is shown in the accompanying figure. What should be her next step? (1) Place the pointer of her compass at X, and make an arc intersecting Q. (2) Place the pointer of her compass at Y, and make an arc intersecting Q. (3) Make a pair of intersecting arcs by placing the pointer of her compass first at A and then at B. (4) Make a pair of intersecting arcs by placing the pointer of her compass first at X and then at Y. 2. In the construction shown in the accompanying figure, which of the following must be true? (1) HM ≌ ME (2) WM ≌ MV (3) WL ≌ WP (4) WV ≌ HE 3. In the construction of a hexagon shown in the accompanying figure, side was constructed first. How was length SN determined? (1) SN equals the length of the radius of circle A. (2) SN equals the length of the diameter of circle A. (3) SN equals twice the length of the diameter of circle A. (4) SN equals one-half the length of the radius of circle A. 4. Claire wants to construct a 30° angle. Which of the following methods could she use? (1) Construct a right triangle, and then bisect one of its angles. (2) Construct an equilateral triangle, and then bisect one of its angles. (3) Construct a hexagon, and then bisect one of its angles. (4) Construct a square, and then bisect one of its angles. B. Free Response—show all work or explain your answer completely 5. Construct a line perpendicular to and passing through V. 6. Construct a line perpendicular to and passing through point V on line. 7. Construct a line parallel to through point J not on line. 8. Construct a regular hexagon. 9. Construct an equilateral triangle inscribed in a circle. 10. Construct a square inscribed in a circle. 11. Construct a triangle whose angles measure 30°, 60°, and 90°. 12. Construct a rectangle whose length and width are in a ratio of 2: 1. 3.3 POINTS OF CONCURRENCY, INSCRIBED FIGURES, AND CIRCUMSCRIBED FIGURES K I Angle bisectors, medians, altitudes, and perpendicular bisectors in triangles have the special property of being concurrent, which means they intersect at a single point. VOCABULARY Some new vocabulary is needed to discuss points of currency in triangles: Concurrent lines—3 or more lines that intersect at a single point. Tangent—a line coplanar with a figure that intersects it at exactly one point, or a segment or ray that lies on a tangent line. *(We also use tangent as an adjective, as in is tangent to circle P.) Figure 3.1 shows tangent to circle P. Figure 3.1 is tangent to circle P at Q. is not a tangent. Point of tangency—the point at which a tangent intersects a circle. Inscribed circle (or incircle)—a circle that is tangent to each of the sides of a polygon. Circumscribed circle (or circumcircle)—a circle that intersects each of the vertices of a polygon. INCENTER The three angle bisectors of any triangle are concurrent at a point called the incenter. The incenter is the center of the circle tangent to each of the three sides of the triangle. This circle is called an inscribed circle. To construct the incenter of any triangle, simply construct the angle bisectors of at least two of the angles. Constructing the third would be a check since the first two will always intersect. If the third angle bisector intersects the first at the same point, then you know the construction is correct. Figure 3.2 shows the construction of the incenter of ΔABC. Figure 3.2 Construction of the incenter of a triangle Once the incenter is located, the point of tangency of the inscribed circle is located by constructing a line from the incenter perpendicular to one of the sides of the triangle. The theorem at work here is that a radius to a point on the circle is perpendicular to the tangent at that point. A circle can then be constructed using the incenter as the center and radius from the incenter to the point of tangency. Figure 3.3 shows the construction of the inscribed circle using the incenter. Figure 3.3 Construction of the inscribed circle using the incenter Theorem—Radius to a Point of Tangency A radius to a point on a circle is perpendicular to the tangent at that point. Since the incenter lies on the three angle bisectors, it is equidistant from the three sides of the triangle. The distance to each of the sides of the triangle is equal to the radius of the circle. The inscribed circle is also the largest circle that can fit inside a triangle, and the incenter is always inside the circle. M F Every triangle can have an inscribed circle, but not every polygon with four or more sides can have one. The incenter of a triangle is equidistant from the three sides of the triangle. CIRCUMCENTER The three perpendicular bisectors of the three sides of any triangle are concurrent at a point called the circumcenter. The circumcenter is also the center of the circle that passes through each of the three vertices of the triangle. This circle is called a circumscribed circle. To construct the circumcenter, construct at least two of the perpendicular bisectors and find the point of concurrency. The circumscribed circle is centered at the circumcenter. The distance to any vertex of the triangle is the radius, as shown in Figure 3.4. Figure 3.4 Construction of the circumcenter and circumscribed circle The theorem at work here is that the points equidistant from two given points lie on the perpendicular bisector. The circumcenter is equidistant from each of the three vertices of triangle, so it must be the center of the circle containing those points. CENTROID The three medians of a triangle are concurrent at a point called the centroid. Each median is constructed by first constructing the midpoint of each side of the triangle, using the perpendicular bisector construction. Then segments are drawn from each vertex to the midpoint of the opposite side. The centroid is the intersection of the three medians. Figure 3.5 shows the construction of centroid P and medians , , and in ΔABC. The centroid of any triangle will always be located inside the triangle. Figure 3.5 Construction of the median of a triangle M F The centroid of a triangle is also the triangle’s center of gravity. The center of gravity is the point where the figure or object is perfectly balanced. ORTHOCENTER The three altitudes of any triangle are concurrent at a point called the orthocenter. Figure 3.6 shows the construction orthocenter P of ΔABC. The orthocenter will be inside of an acute triangle, on a vertex of a right triangle, and outside an obtuse triangle. Figure 3.6 Construction of the orthocenter, P, of triangle ΔABC POINTS OF CONCURRENCY IN TRIANGLES Segment Where Concurrent Feature Segment Where Concurrent Feature Perpendicular Circumcenter Center of the circumscribed circle; bisector equidistant from each of the three vertices Angle bisector Incenter Center of the inscribed circle; equidistant from the three sides of the triangle Median Centroid Divides each median in a 2 : 1 ratio; is the center of gravity Altitude Orthocenter Located inside acute triangles, on a vertex of right triangles, and outside obtuse triangles M T Mnemonic devices can be helpful for memorizing information. An example for points of concurrency is “all of my children are bringing in peanut butter cookies.” AO—altitudes/orthocenter MC—medians/centroid ABI—angle bisectors/incenter PBC—perpendicular bisectors/circumcenter Check Your Understanding of Section 3.3 Multiple-Choice 1. A phone company wants to locate a cell phone tower equidistant from three cities. An appropriate strategy to find the correct location would be to construct a

Let's Review Regents: Geometry 2020 PDF

Document Details

Tags

Related

Summary

Full Transcript