Solution Manual For Elementary Geometry for College Students, 7th Edition

Complete Solutions Manual Elementary Geometry for College Students SEVENTH EDITION Daniel Alexander Parkland College, Professor Emeritus Geralyn M. Koeberlein Mahomet-Seymour High School, Mathematics Department Chair, Retired Prepared by Geralyn M. Koeberlein Mahomet-Seymour High School, Mathematics Department Chair, Retired Daniel Alexander Parkland College, Professor Emeritus Australia • Brazil • Mexico • Singapore • United Kingdom • United States © 2020 Cengage Learning ISBN: 978-0-357-02221-4 Unless otherwise noted, all content is © Cengage Cengage 20 Channel Center Street Boston, MA 02210 USA ALL RIGHTS RESERVED. No part of this work covered by the copyright herein may be reproduced or distributed in any form or by any means, except as permitted by U.S. copyright law, without the prior written permission of the copyright holder. For product information and technology assistance, contact us at Cengage Customer & Sales Support, 1-800-354-9706 or support.cengage.com. 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The Supplement is for your personal, noncommercial use only and may not be reproduced, posted electronically or distributed, except that portions of the Supplement may be provided to your students IN PRINT FORM ONLY in connection with your instruction of the Course, so long as such students are advised that they may not copy or distribute any portion of the Supplement to any third party. You may not sell, license, auction, or otherwise redistribute the Supplement in any form. We ask that you take reasonable steps to protect the Supplement from unauthorized use, reproduction, or distribution. Your use of the Supplement indicates your acceptance of the conditions set forth in this Agreement. If you do not accept these conditions, you must return the Supplement unused within 30 days of receipt. All rights (including without limitation, copyrights, patents, and trade secrets) in the Supplement are and will remain the sole and exclusive property of Cengage Learning and/or its licensors. The Supplement is furnished by Cengage Learning on an “as is” basis without any warranties, express or implied. This Agreement will be governed by and construed pursuant to the laws of the State of New York, without regard to such State’s conflict of law rules. Thank you for your assistance in helping to safeguard the integrity of the content contained in this Supplement. We trust you find the Supplement a useful teaching tool. Contents Suggestions for Course Design iv Chapter-by-Chapter Commentary v Solutions Chapter P Preliminary Concepts 1 Chapter 1 Line and Angle Relationships 5 Chapter 2 Parallel Lines 24 Chapter 3 Triangles 51 Chapter 4 Quadrilaterals 75 Chapter 5 Similar Triangles 103 Chapter 6 Circles 137 Chapter 7 Locus and Concurrence 160 Chapter 8 Areas of Polygons and Circles 177 Chapter 9 Surfaces and Solids 210 Chapter 10 Analytic Geometry 233 Chapter 11 Introduction to Trigonometry 276 Appendix A Algebra Review 297 iii © 2020 Cengage Learning, Inc. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Suggestions for Course Design The authors believe that this textbook would be appropriate for a 3-hour, 4-hour, or 5-hour course. Some instructors may choose to include all or part of Appendix A (Algebra Review) due to their students’ background in algebra. There may also be a desire to include The Introduction to Logic, found at our website, as a portion of the course requirement. Inclusion of some laboratory work with a geometry package such as Geometry Sketchpad is an option for course work. 3-hour course Include most of Chapters 1–8. Optional sections could include: Section 2.2 Indirect Proof Section 2.3 Proving Lines Parallel Section 2.6 Symmetry and Transformations Section 3.5 Inequalities in a Triangle Section 6.4 Some Constructions and Inequalities for the Circle Section 8.5 More Area Relationships in the Circle 4-hour course Include most of Chapters 1–8 and include all/part of at least one of these chapters: Chapter 9 Surfaces and Solids (Solid Geometry) Chapter 10 Analytic Geometry (Coordinate Geometry) Chapter 11 Introduction to Trigonometry 5-hour course Include most of Chapters 1–11 as well as topics desired from Appendix A and/or The Introduction to Logic (see website). Daniel C. Alexander Geralyn M. Koeberlein iii © 2020 Cengage Learning, Inc. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Chapter-by-Chapter Commentary for Instructors Chapter P: Preliminary Concepts Section P.1: Sets and Sets of Numbers In this section, students review the notions and basic terms related to sets of objects. Given a set or description of a set, the student should be able to classify that set as empty, finite, or infinite. For the path provided by a set of points, the student should be able to characterize the path as continuous or discontinuous and also to describe that path as straight, curved, circular, or scattered. The student should recognize certain subsets of a straight line as a line segment or ray. Given two sets, the student should be able to form their union or their intersection. In turn, students should utilize Venn diagrams to display two sets that are disjoint or the union or intersection of the two sets. Section P.2: Statements and Reasoning The student should realize that statements of geometry appear in both words or symbols and can be classified as true or false. Of the compound statements (conjunction, disjunction, and implication), the instructor should warn the student of the significance of the implication in that it (the “If . . ., then . . .” statement) is most relevant in deductive reasoning. For the implication (also known as a conditional statement), the student should be able to determine its hypothesis and conclusion; this determination acts as an important prerequisite for preparing a proof. Also, the student should be able to recognize and distinguish the three type of reasoning (intuition, induction, and deduction). Further, the Law of Detachment plays a major role in the development/advancement of geometry. Emphasizing that valid arguments can be confused with invalid arguments will alert students to potential pitfalls. Section P.3: Informal Geometry and Measurement In this section, many terms of geometry are introduced informally; in Chapter 1, these vocabulary terms will be presented formally. For students who seem to be poorly prepared, this approach (both an informal and a formal introduction to geometric terminology) may prove quite helpful. Measuring the line segment’s length with a ruler prepares the student intuitively for the Ruler Postulate and the Segment Addition Postulate of Chapter 1; similarly, measuring angles with a protractor also prepares the student with the insights needed to deal with topics found in Section 1.2. Students that have difficulty with measures of angles (likely due to the dual scales found on protractors) can correct this situation by considering an activity sheet which focuses upon measuring angles with a protractor. Chapter One: Line and Angle Relationships Section 1.1: Early Definitions and Postulates So that the student can understand the concept “branch of mathematics,” he or she should be introduced to the four parts of a mathematical system. The basic terminology and symbolism for lines (and their subsets) must be given due attention because these will be v © 2020 Cengage Learning, Inc. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. utilized throughout the textbook. The instructor should alert students to undefined terms such as “building blocks.” Also, characterize definitions and postulates as significant in that they lead to conclusions known as theorems, or statements that can be proven. For the instructor, pens and pencils can be used to visualize relationships among lines, line segments, and rays. Table tops and pieces of cardboard can be used to represent planes. Section 1.2: Angles and Their Relationships It is most important, once again, that students be able to not only recognize the terminology for angles, but also to be able to state definitions and principles in their own terms. Measuring angles with the protractor should enable the student to understand principles such as the Protractor Postulate and the Angle Addition Postulate. Constructions may also provide understanding of certain concepts (like congruence and angle bisector). Many examples will remind the student of algebra’s role in the solution of problems of geometry. Students can be referred to the Algebra Review (Appendix A) as needed. Section 1.3: Introduction to Geometric Proof The purpose of this section is to introduce the student to geometric proof. Many of the little things (hypothesis = given information, order, statements and reasons, etc.) are of tremendous importance as you prepare the student for proof. In the Sixth Edition, many of the techniques are emphasized in the feature Strategy for Proof; be sure that your students are aware of this feature and utilize these techniques. The two-column proof is used at this time because it emphasizes all the written elements of proof. Section 1.4: Relationships: Perpendicular Lines The “perpendicular relationship” is most important to many later discoveries. For now, be sure that students know that this relation extends itself to combinations such as line-line, line-plane, and plane-plane. For the general concept of relation, we explore the reflexive, symmetric, and transitive properties––particularly those that relate geometric figures. Some discussion of uniqueness is productive in that it will provide background for the notion of auxiliary lines (introduced in a later section). Section 1.5: The Formal Proof of a Theorem Be sure that your students know in order the five written parts of the written proof: Statement of proof (the theorem), drawing (from hypothesis), given (from hypothesis), prove (from conclusion), and proof. The instructor must help the student understand that the unwritten Plan for Proof is far and away the most important step; for this part, suggest scratch paper, reviewing the textbook, and use of the Strategy for Proof feature. Several theorems that have already been stated or proven in part are left as exercises; many of these have a similar counterpart (an example) in the textbook. Chapter Two: Parallel Lines Section 2.1: The Parallel Postulate and Special Angles From the outset of Chapter 2, the instructor should emphasize that parallel lines must be coplanar. It is suggested that the instructor illustrate parallel and perpendicular (even skew lines) relationships by using pens and pencils for lines and pieces of construction vi © 2020 Cengage Learning, Inc. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. paper or cardboard for planes. Even though it is nearly impossible for students to grasp the significance of this fact, tell students that the Parallel Postulate characterizes the branch of mathematics known as Euclidean Geometry (plane geometry). While this characterization suggests that “the Earth is flat,” it is adequate for our study even though spherical geometry is required at the global level. Beginning with Postulate 11, students should be able to complete several statements of the form, “If two parallel lines are cut by a transversal, then . . . .” Section 2.2: Indirect Proof Note: If there is insufficient time allowed for the complete development of geometry from a theoretical perspective, this section can be treated as optional. This section provides the opportunity to review the negation of a statement as well as the implication and its related statements (converse, inverse, and contrapositive). Based upon the deductive form Law of Negative Inference, the primary goal of this section is the introduction of the indirect proof. It is important that students be aware that the indirect proof is often used in proving negations and uniqueness theorems. In the construction of an indirect proof, the student often makes the mistake of assuming that the negation of the hypothesis (rather than negation of conclusion) is true. Section 2.3: Proving Lines Parallel Due to the similarity among statements of this section and those in Section 2.1, caution students that parallel lines were a given in Section 2.1. However, theorems in Section 2.3 prove that lines meeting specified conditions are parallel; that is, statements in this section take the form, “If . . . , then these lines are parallel.” For this section, have students draw up a list of conditions that lead to parallel lines. Section 2.4: The Angles of a Triangle Students will need to become familiar with much of the terminology of triangles (sides, angles, vertices, etc.). Also, students should classify triangles by using both side relationships (scalene, isosceles, etc.) and angle relationships (obtuse, right, etc.). Some persuasion may be needed to have students accept the use of an auxiliary line. For an auxiliary line, you must (1) explain its uniqueness, (2) verify its existence in a proof, and (3) explain why that particular line was chosen. The instructor cannot emphasize enough the role of the theorem, “The sum of the measures of the interior angles of a triangle is 180°.” Because of the relation of remaining theorems to Theorem 2.4.1, note that each statement is called a corollary of that theorem. Section 2.5: Convex Polygons Again, terminology for the polygon must be given due attention. The student should be able to classify several polygons due to the number of sides (triangle, quadrilateral, pentagon, etc.). Terms such as equilateral, equiangular, and regular should be known. Rather than count the number of diagonals D for a polygon of n sides, the student should n(n − 3) be able to use the formula D = . The student should be able to state and use 2 formulas for the sum of the interior angles (or exterior angles) of a polygon; in turn, the vii © 2020 Cengage Learning, Inc. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. student should know and be able to apply formulas that lead to the measures of an interior angle or exterior angle of a regular polygon. Polygrams can be treated as optional. Section 2.6: Symmetry and Transformations Appealing to the student’s intuitive sense of symmetry, the student can be taught that line symmetry exists when one-half of the figure is the mirror image (reflection) of the other half, with the line of symmetry as the mirror. For point symmetry, ask the student “is there a point (not necessarily on the figure) that is the midpoint of a line segment determined by two corresponding points on the figure in question.” While a figure may have more than one line of symmetry, emphasize that the figure can have only one point of symmetry. Transformations (slides, reflections, and rotations) always produce an image (figure) that is congruent to the original figure. In Chapter 3, many examples of pairs of congruent triangles can be interpreted as the result of a slide, reflection, or rotation of one triangle to produce another triangle (its image). Chapter Three: Triangles Section 3.1: Congruent Triangles As you begin the study of congruent triangles, stress the need to pair corresponding vertices, corresponding sides, and corresponding angles. Also, students should realize that the methods for proving triangles congruent (SSS, SAS, ASA, and AAS) are useful throughout the remainder of their study of geometry. Due to the simplicity and brevity of some proof problems found in this section, encourage students to attempt proof without fear. Also, have students utilize suggestions found in the Strategy for Proof feature. Section 3.2: Corresponding Parts of Congruent Triangles Students should know the acronym CPCTC and know that it represents, “Corresponding Parts of Congruent Triangles are Congruent.” Emphasize that CPCTC allows them to prove that a pair of line segments (or a pair of angles) are congruent; however, warn them that CPCTC cannot be cited as a reason unless a pair of congruent triangles have already been established. Let students know that CPCTC empowers them to take an additional step; for instance, proving that 2 line segments are congruent may enable the student to establish a midpoint relationship. Once terminology for the right triangle has been introduced, caution students that the HL method for proving triangles congruent is valid only for right triangles. In order to give it due attention, the Pythagorean Theorem is introduced here without proof. The connection of the Pythagorean Theorem to this section lies in the fact that it will later be used to prove the HL theorem. Section 3.3: Isosceles Triangles Students should become familiar with terms (base, legs, etc.) that characterize the isosceles triangle. Students should know meanings of (and be able to differentiate between) these figures related to a triangle: an angle-bisector, the perpendicular-bisector of a side, an altitude, and a median. Of course, every triangle will have three anglebisectors, three altitudes, etc. With unsuspecting students, it may be best to show them that the three-perpendicular bisectors of sides (or three altitudes) can intersect at a point outside the triangle; perhaps a drawing session would help! The most important theorems of this section are converses: (1) If two sides of a triangle are congruent, then the angles viii © 2020 Cengage Learning, Inc. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. opposite these sides are congruent, and (2) If two angles of a triangle are congruent, then the sides opposite these angles are congruent. Section 3.4: Basic Constructions Justified Note: If there is insufficient time or constructions are not to be emphasized, this section can be treated as optional. The first goal of this section is to validate (prove) the construction methods introduced in earlier sections. For instance, we validate the method for bisecting an angle through the use of congruent triangles and CPCTC. The second goal of this section is that of constructing line segments of a particular length or of constructing angles of a particular measure (such as 45° or 60°). Section 3.5: Inequalities in a Triangle Note: If there is insufficient time or inequality relationships are not to be emphasized, this section can be treated as optional. To enable the proofs of theorems in this section, we must begin with a concrete definition of the term greater than. Note that some theorems involving inequalities are referred to as lemmas because these theorems help us to prove other theorems. The inequality theorems involving the lengths of sides and measures of angles of a triangle are very important because they will be applied in Chapters 4 and 6. For some students, the Triangle Inequality will later be applied in the coursework of trigonometry and calculus. Chapter Four: Quadrilaterals Section 4.1: Properties of a Parallelogram Alert students to the fact that principles of parallel lines, perpendicular lines, and congruent triangles are extremely helpful in developments of this chapter. Be sure to define the parallelogram, but caution students not to confuse this definition with any of several properties of parallelograms found in theorems of this section. These theorems have the form, “If a quadrilateral is a parallelogram, then . . . .” In Section 4.3, these properties will also characterize the rectangle, square, and rhombus, because each is actually a special type of parallelogram. The final topic (bearing of airplane or ship) can be treated as optional. Section 4.2: The Parallelogram and Kite In this section, parallelograms are not a given in the theorems of the form, “If a quadrilateral . . . , then the quadrilateral is a parallelogram.” That is, we will be proving that certain quadrilaterals are parallelograms. Like the parallelogram, a kite has two pairs of congruent sides; by definition, the congruent pairs of sides in the kite are adjacent sides. A kite has its own properties (like perpendicular diagonals) as well. Section 4.3: The Rectangle, Square, and Rhombus Consider carefully the definition of each figure (rectangle, square, and rhombus); with each being a type of parallelogram, the properties of parallelograms are also those of the rectangle, square, and rhombus. Of course, each type of parallelogram found in this ix © 2020 Cengage Learning, Inc. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. section has its own properties. For example, the rectangle and square have four right angles while the diagonals of a rhombus are perpendicular; as a consequence of these properties, the Pythagorean Theorem can be applied toward solving many problems involving these special types of parallelograms. Section 4.4: The Trapezoid Because the trapezoid has only two sides that are parallel, it does not assume the properties of parallelograms. If the trapezoid is isosceles, then it will have special properties such as congruent diagonals and congruent base angles. Remaining theorems describe the length of a median of a trapezoid and characterize certain quadrilaterals as trapezoids or isosceles trapezoids. Quadrilateral types can be compared by use of a Venn diagram or the following outline: 1. Quadrilaterals A. Parallelograms 1. Rectangle a. Square 2. Rhombus B. Kites C. Trapezoids 1. Isosceles Trapezoids Chapter Five: Similar Triangles Section 5.1: Ratios, Rates, and Proportions Note: For work in Chapter 5, the instructor may want to refer those students who need a review of the methods of solving quadratic equations to Appendix Sections A.4 and A.5. In this section, emphasize the difference between a ratio (quotient comparing like units) and a rate (quotient comparing unlike units). Students should understand that a proportion is an equation in which two ratios (or rates) are equal. Terminology for proportions (means, extremes, etc.) are important because the student better understands a property like the Means-Extremes Property. Section 5.2: Similar Polygons In this section, similar polygons are defined and their related terminology (corresponding sides, corresponding angles, etc.) are introduced. The definition of similar polygons allows students to (1) equate measures of corresponding angles, and (2) form proportions that compare lengths of corresponding sides. Thus, this section focuses on problem solving strategies, including an ancient technique known as shadow reckoning. Section 5.3: Proving Triangles Similar Whereas Section 5.2 focuses on problem solving, Section 5.3 emphasizes methods for proving that triangles are similar. Due to its simplicity, the instructor should emphasize that the AA method for proving triangles similar should be used whenever possible. The definition of similar triangles forces two relationships among parts of similar triangles: (1.) CASTC means “Corresponding Angles of Similar Triangles are Congruent,” while (2.) CSSTP means “Corresponding Sides of Similar Triangles are Proportional.” x © 2020 Cengage Learning, Inc. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Other methods for proving triangles similar are SAS and SSS; in application, these methods are difficult due to the necessity of showing lengths of sides to be proportional. Warn students not to use SAS and SSS (methods of proving triangles congruent) as reasons for claiming that triangles are similar. Section 5.4: The Pythagorean Theorem Theorem 5.3.1 leads to a proof of the Pythagorean Theorem and its converse. Students should be aware that many (more than 100) proofs exist for the Pythagorean Theorem. For emphasis, note that the Pythagorean Theorem allows one to find the length of a side of a right triangle; however, its converse enables one to conclude that a given triangle may be a right triangle. Because these are commonly applied, Pythagorean Triples such as (3,4,5) and (5,12,13) are best memorized by the student. With c being the length of the longest side of a given triangle, this triangle is: an acute triangle if c 2 a 2 + b 2 . Section 5.5: Special Right Triangles In Section 5.4, some right triangles were special because of their integral lengths of sides (a,b,c). In Section 5.5, a right triangle with angle measures of 45°, 45°, and 90° always has congruent legs while the hypotenuse is 2 times as long as either leg. Also, a right triangle with angle measures of 30°, 60°, and 90° has a longer leg that is 3 times as long as the shorter leg, while the hypotenuse is two times as long as the shorter leg. These relationships, and their converses, also have applications in trigonometry and calculus. Section 5.6: Segments Divided Proportionally Note: In this section, Ceva’s Theorem is optional in that it is not applied in later sections. The phrase divided proportionally can be compared to profit sharing among unequal partners in a business venture. This concept is, of course, the essence of numerous applications found in this section. The Angle-Bisector Theorem states that an anglebisector of an angle in a triangle leads to equal ratios among the parts of the lengths of the two sides forming the bisected angle and the lengths of parts of the third side. Chapter Six: Circles Section 6.1: Circles and Related Segments and Angles Terminology for the circle is reviewed and extended in this section. Students will have difficulty with the definition of congruent arcs in that they must have both equal measures and lie within the same circle or congruent circles. Many of the principles of this section are intuitive and therefore easily accepted. Contrast the sides and vertex locations of the central angle and the inscribed angle. Stress these angle-measurement relationships in that further angle-measurement relationships will be added in Section 6.2. Section 6.2: More Angle Measures in the Circle The terms tangent and secant are introduced and will be given further attention in later sections as well as in the coursework of trigonometry and calculus. Again emphasize the xi © 2020 Cengage Learning, Inc. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. new angle-measurement techniques with the circle. A summary of methods (Table 6.1) is provided for students. Section 6.3: Line and Segment Relationships in the Circle The early theorems in this section sound similar, yet make different assertions; for this reason, it may be best that students draw the hypothesis of each theorem to “see” that the conclusion must follow. Students will also need to distinguish the concepts of common tangent for two circles and tangent circles. Each of the relationships found in Theorems 6.3.5–6.3.7 is difficult to believe without proof; however, with the help of an auxiliary line, each proof of theorem is easily and quickly proved. Section 6.4: Some Constructions and Inequalities for the Circle Note: If there is insufficient time or constructions and inequality relationships are not to be emphasized, this section can be treated as optional. Because the construction methods of this section are fairly involved, be sure to assign homework exercises that have students perform them. The inequality relationships involving circles are intuitive (easily believed); due to the difficulty found in constructing proofs of these theorems, the instructor may wish to treat proofs as optional. Chapter Seven: Locus and Concurrence Section 7.1: Locus of Points So that the term locus is less confusing for students, the instructor should tie this word to its Latin meaning: “location.” For the locus concept, quantity makes a difference; that is, students will need to see several examples. While construction of a locus is optional, a drawing of the locus is imperative. Theorems 7.1.1 and 7.1.2 are most important in that they lay the groundwork for later sections. The instructor should be sure to distinguish between the locus of points in a plane and the locus of points in space. Section 7.2: Concurrence of Lines The discussion of locus leads indirectly to the notion of concurrence. In particular, the concurrence of the three angle-bisectors of a triangle follows directly from the first locus theorem in Section 7.1; in turn, a triangle has an inscribed circle whose center is the incenter of the triangle. Likewise, the three perpendicular-bisectors of the sides of a triangle are concurrent at the circumcenter of the triangle, the point that is the center of the circumscribed circle of every triangle. In this section, not only have students memorize the terms incenter, circumcenter, orthocenter, and centroid, but also have them know which concurrency (angle-bisectors, etc.) leads to each result. Section 7.3: More About Regular Polygons Based upon our findings in Section 7.2, the student should know that a circle can be inscribed in every triangle and also be circumscribed about every triangle. Further, the center for both circles (inscribed and circumscribed) is the same point for the equilateral triangle and regular polygons in general. The new terminology for the regular polygon (center, radius, apothem, central angle, etc) should be memorized because it will also be applied in Section 8.3. xii © 2020 Cengage Learning, Inc. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Chapter Eight: Areas of Polygons and Circles Section 8.1: Area and Initial Postulates Even though most students say area of a triangle, they should realize that the more accurate description would be area of triangular region. Stress the difference between linear units (used to measure length) and square units (used to measure area). With each area formula serving as a “stepping stone” to the next formula, the given order for the area formulas is natural. Perhaps the most significant formula in the list is that of the parallelogram (A = bh) in that this is derived from the area of rectangle formula while it leads to the remaining formulas. Section 8.2: Perimeter and Area of Polygons Given its practical applications, the notion of perimeter should be reviewed and extended. Heron’s Formula is difficult to state and apply; however, it is common to find the area of a triangle whose lengths of sides are known. The proof of Heron’s Formula is found at the website that accompanies this textbook. Emphasize Theorem 8.2.3 and that the area formulas for the rhombus and kite are just special cases of this theorem. Section 8.3: Regular Polygons and Area In this section, we first consider formulas for the area of the equilateral triangle and square. For regular polygons in general, be sure to introduce or review the terminology (center, radius, apothem, central angle, etc.) that was found in Section 7.3; if studied, the work in both Chapters 9 and 11 use this terminology as well. The ultimate goal of this 1 section is to establish the formula for the area of a regular polygon, namely A = aP. 2 Section 8.4: Circumference and Area of a Circle Begin with the definition of π as a ratio and then provide some approximations of its 22 C and 3.1416). Using π = , we can show that C = π d and C = 2 π r. value (such as 7 d Using a proportion, we find the length of an arc of circle (as part of the circumference). Developed as the limit of areas of inscribed regular polygons, we show that the area of a circle is given by A = π r 2 . Note that the concept of limit needs a few examples. For students to distinguish between 2 π r and π r 2 (for circumference and area), have them compare units, where r = 3 cm, 2 π r = 2 π ⋅ 3 cm = 6 π cm (a linear measure) while π r 2 = π ⋅ 3 cm ⋅ 3 cm or 9 π cm 2 (a measure of area). Section 8.5: More Area Relationships in the Circle Note: If there is insufficient time for the study of this section, it can be treated as optional in that none of the content is used in later sections. Formulas for the area of a sector and segment depend upon the formula for the area of a circle; however, an understanding of these area concepts is far more important than the memorization of formulas. The area of segment applications require that the related central angle have a convenient measure, such as 60°, 90°, or 120°; otherwise, xiii © 2020 Cengage Learning, Inc. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. trigonometry would be necessary to solve the problem. When a triangle has perimeter P 1 and inscribed circle of radius r, the area of the triangle is given by A = rP. 2 Chapter Nine: Surfaces and Solids Section 9.1: Prisms, Area, and Volume The student should consider three-dimensional objects in this section and chapter; for that purpose, the instructor should use a set of models displaying various prisms and other solids or space figures. Students need to become familiar with prisms and related terminology. To calculate the lateral area and the total area of a prism, a student must apply formulas from Chapter 8. For the volume formula for a prism (V = Bh), emphasize that B is the area of the base and that V is always measured in cubic units. Section 9.2: Pyramids, Area, and Volume Again, the instructor should use a set of models to display various pyramids. Students need to become familiar with pyramids and related terminology, including the slant height of a regular pyramid. Calculating the lateral area and the total area of a pyramid requires the application of formulas from Chapter 8. To find the length of the slant height of a regular pyramid requires the use of the Pythagorean Theorem. Compare the formula 1 for the volume of a pyramid (V = Bh) to that of the prism (V = Bh). 3 Section 9.3: Cylinders and Cones Comparing the prism to cylinder and the pyramid to cone will help to motivate students in learning the area and volume formulas of this section. Three-dimensional models will motivate the formula for the lateral area of cylinder and to explain the slant height of the right circular cone. The length of the slant height of the right circular cone can be found by using the Pythagorean Theorem. Compare volume formulas for the prism (V = Bh) and right circular cylinder (V = Bh or V = π r 2 h); likewise compare the volume formulas 1 1 1 for the pyramid (V = Bh) and right circular cylinder (V = Bh or V = π r 2 h). While 3 3 3 the material involving solids of revolution is a preparatory topic for calculus, it can be treated as optional. Section 9.4: Polyhedrons and Spheres Students should recognize (or be told) that prisms and pyramids are merely examples of polyhedrons (or polyhedra). Students should verify Euler’s Formula (V + F = E + 2) for polyhedra with a small number of vertices by using solid models from a kit. For the sphere, compare its terminology with that of the circle; however, note that a sphere also has tangent planes. To develop the volume of sphere formula, it is necessary to interpret the volume as the limit of the volumes of inscribed regular polyhedra with an increasing number of faces. Due to limitations, we only apply the surface area of sphere formula. xiv © 2020 Cengage Learning, Inc. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Chapter Ten: Analytic Geometry Section 10.1: The Rectangular Coordinate System The student should become familiar with the rectangular coordinate system and its related terminology. Warn students that the definitions for lengths of horizontal and vertical line segments are fairly important in the development of the chapter. For the formulas developed, P 1 is read first point and x 1 as the value of x for the first point. The Distance Formula and Midpoint Formulas are must be memorized in that they will be used throughout Chapter 10; of course, these formulas are also useful in later coursework as well. Section 10.2: Graphs of Linear Equations and Slope At first, a point-plot approach for graphing equations is used. However, graphing linear equations leads to graphs that are lines and, in turn, the notion of slope of a line. The student must memorize the Slope Formula. By sight, a student should be able to recognize that a given line has a positive, negative, zero, or undefined slope. Many students have difficulty drawing a line based upon its provided slope; for this rise . To draw a line with slope m, move from reason, it is important to treat slope as m = run one point to the second point by simultaneously using a vertical change (rise) that corresponds to the horizontal change (run). Using the slopes of two given lines, the student should be able to classify lines as parallel, perpendicular, or neither. Section 10.3: Preparing to Do Analytic Proofs This section is a “warm up” for completing analytic proofs that follow in Section 10.4. Specific goals that need to be achieved are: 1. The student should know the formulas found in the summary on the first page. 2. The student should follow the suggestions for placement of a drawing so that the proof of the theorem can be completed. See the Strategy for Proof. 3. The student should study the relationship between desired theorem conclusions and formulas needed to obtain such conclusions. See the Strategy for Proof. Section 10.4: Analytic Proofs This section utilizes all formulas and suggestions from previous sections of Chapter 10. In each classroom, the instructor must warn students of the amount of rigor required. For instance, suppose that we are trying to prove a theorem such as, “If a quadrilateral is a parallelogram, then its diagonals bisect each other.” Does the student provide a figure with certain vertices that is known to be a parallelogram, or does that figure have to be proven a parallelogram before the proof can be continued? You may wish to prove each claim once and then accept it at a later time as given (not needing proof); if it was shown in an earlier section that the triangle with vertices at A(−a,0), B(a,0), and C(0,b) is isosceles, then it will be given as such in a later proof. Section 10.5: Equations of Lines In this section, we use given information about a line (like slope and y-intercept) to find its equation. Students will need to memorize and apply both the Slope-Intercept and the Point-Slope forms of a line. Emphasize that solving systems of linear equations is the xv © 2020 Cengage Learning, Inc. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. algebraic equivalent of finding the point of intersection of two lines using geometry. Emphasize that the method (algebra or geometry) used to find this point of intersection always leads to the same result. Point out that the Slope-Intercept and Point-Slope forms of a line can be used to prove further theorems by the analytic approach. Section 10.6: The Three-Dimensional Coordinate System In this section, we plot points of the form (x,y,z) in three dimensions. Warn students that the forms of equations of a line will seem unfamiliar; however, the equation of a plane in Cartesian space is similar to the general form for the equation of line in the Cartesian plane. Students should easily adapt to the natural extensions of the Distance Formula and Midpoint Formula. Ironically, there is no Slope Formula. For the concept of direction vector, the student will need some convincing of its importance; however, the direction vectors for two lines will determine whether these lines have the same direction (parallel or coincident) or different directions (intersecting or skew). To consider the relationships between planes, the instructor will need to give considerable attention to algebraic techniques in that the methods will be more involved. This section concludes with the equation of a sphere in Cartesian space, again seen by students as a natural extension of the equation of a circle in the Cartesian plane. Chapter Eleven: Introduction to Trigonometry Section 11.1: The Sine Ratio and Applications Related to the right triangle, ask students to memorize the sine ratio of an angle in the opposite form ; while this seems rather informal, the remaining definitions of hypotenuse trigonometric ratios will be given in a similar form. While students are encouraged to use the calculator to find sine ratios for angles, they should also know these results from 1 2 3 , sin 60° = , and sin 90° = 1. memory: sin 0° = 0, sin 30° = , sin 45° = 2 2 2 Students should realize that the sine ratios increase as the angle measure increases. Emphasize the terms angle of elevation and angle of depression and be able to perform applications that require the use of the sine ratio. Section 11.2: The Cosine Ratio and Applications adjacent . hypotenuse In addition to using the calculator to find cosine ratios, students should memorize results 3 such as cos 0° = 1, cos 30° = , etc. Students should recognize that an increase in 2 angle measures produces a decrease in cosine measures. Students need to be able to complete applications that require the cosine ratio. The instructor should include and perhaps require that the student be able to prove the theorem sin 2 θ + cos 2 θ = 1. Emphasize that many geometry problems (such as Example 7 of this section) cannot be solved without the use of trigonometry. Ask students to memorize the cosine ratio of an angle in the form xvi © 2020 Cengage Learning, Inc. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Section 11.3: The Tangent Ratio and Other Ratios opposite Students should memorize the tangent ratio as and memorize exact values for adjacent tan 0°, tan 30°, tan 45°, etc. Some attention and discussion should be devoted to the claim that “tan 90° is undefined.” Now that three ratios are available, some practice and discussion should be given to determination of the ratio needed to solve a particular problem. While the remaining ratios (cotangent, secant, and cosecant) are included for completeness, the students can solve all problems by using only the sine, cosine, and tangent ratios. The final ratios can be recalled as reciprocals of the first three; for a b instance, if sin θ = , then csc θ = . b a Section 11.4: Applications with Acute Triangles Only the most basic trigonometric identities are included in this section. Due to the Reciprocal Identities, remind students that only the sine, cosine, and tangent ratios are needed in application. The instructor should demonstrate the use of the calculator in finding a ratio such as sec 34° (as reciprocal of cos 34°). Because the Quotient Identities and Pythagorean Identities are easily proved, some time should be devoted to proving at 1 least one identity of each type. The area formula, A = bc sin α , is easily proved; 2 however, students should focus on its application. Students should also know the general form of the the Law of Sines and the Law of Cosines; also, the student should be able to determine which form of each is used to solve a problem. In this textbook, we do not touch on ratios, identities, or formulas involving an obtuse angle. xvii © 2020 Cengage Learning, Inc. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Chapter 1 Line and Angle Relationships SECTION 1.1: Early Definitions and Postulates 1. AC 2. Midpoint 3. 6.25 ft ⋅ 12 in./ft = 75 in. 4. 52 in. ÷ 12 in./ft = 4 1 ft or 4 ft 4 in. 3 5. 1 m ⋅ 3.28 ft/m = 1.64 feet 2 6. 16.4 ft ÷ 3.28 ft/m = 5 m 7. 18 – 15 = 3 mi 8. 300 + 450 + 600 = 1350 ft 1350 ft ÷ 15 ft/s = 90 s or 1 min 30 s 9. a. A-C-D b. A, B, C or B, C, D or A, B, D 10. a. Infinite 15. 2 x + 1 = 3x − 2 − x = −3 x=3 AM = 7 16. 2( x + 1) = 3( x − 2) 2 x + 2 = 3x − 6 − 1x = − 8 x =8 AB = AM + MB AB = 18 + 18 = 36 17. 2 x + 1 + 3x + 2 = 6 x − 4 5x + 3 = 6x − 4 −1x = −7 x=7 AB = 38 18. No; Yes; Yes; No JJJG JJJG 19. a. OA and OD JJJG JJJG b. OA and OB (There are other possible answers.) HJJG 20. CD lies on plane X. b. One c. None 21. a. d. None HJJG 11. CD means line CD; CD means segment CD; CD means the measure or length of CD ; JJJG CD means ray CD with endpoint C. b. 12. a. No difference b. No difference c. No difference JJJG d. CD is the ray starting at C and going toward D. JJJG DC is the ray starting at D and going toward C. c. 13. a. m and t b. m and p or p and t 14. a. False b. False c. True d. True e. False © 2020 Cengage Learning. All rights reserved. 5 6 Chapter 1: Line and Angle Relationships 22. a. 28. a. Equal b. Equal c. AC is twice CD. 29. Given: AB and CD as shown (AB > CD) Construct MN on line A so that MN = AB + CD b. 30. Given: AB and CD as shown (AB > CD) Construct: EF on line A so that EF = AB − CD . c. 31. Given: AB as shown Construct: PQ on line n so that PQ = 3( AB ) 32. Given: AB as shown Construct: TV on line n so that TV = 1 ( AB ) 2 HJJG 23. Planes M and N intersect at AB . 24. B 25. A 26. a. One b. Infinite c. One d. None 27. a. C b. C 33. a. No b. Yes c. No d. Yes c. H © 2020 Cengage Learning. All rights reserved. Section 1.2 34. A segment can be divided into 2n congruent parts, where n ≥ 1 . 7 10. a. True b. False 35. Six c. False 36. Four d. False 37. Nothing e. True 38. a. One 11. a. Obtuse b. One b. Straight c. None c. Acute d. One d. Obtuse e. One f. One g. None 39. a. Yes b. Yes 12. B is not in the interior of ∠FAE ; the AngleAddition Postulate does not apply. 13. m∠FAC + m∠CAD = 180 ∠FAC and ∠CAD are supplementary. 14. a. x + y = 180 b. x = y c. No 40. a. Yes 15. a. x + y = 90 b. x = y b. No c. Yes 41. 1 1 2a + 3b a + b or 3 2 6 SECTION 1.2: Angles and Their Relationships 1. a. Acute b. Right c. Obtuse 2. a. Obtuse b. Straight c. Acute 3. a. Complementary b. Supplementary 4. a. Congruent 16. 62° 17. 42° 18. 2 x + 9 + 3x − 2 = 67 5 x + 7 = 67 5 x = 60 x = 12 19. 2 x − 10 + x + 6 = 4( x − 6) 3 x − 4 = 4 x − 24 20 = x x = 20 m∠RSV = 4(20 − 6) = 56° 20. 5( x + 1) − 3 + 4( x − 2) + 3 = 4(2 x + 3) − 7 5 x + 5 − 3 + 4 x − 8 + 3 = 8 x + 12 − 7 9 x − 3 = 8x + 5 x =8 m∠RSV = 4(2 ⋅ 8 + 3) − 7 = 69° 21. x x + = 45 2 4 b. None Multiply by LCD, 4 5. Adjacent 6. Vertical 7. Complementary (also adjacent) 8. Supplementary 9. Yes; No © 2020 Cengage Learning. All rights reserved. 2 x + x = 180 3 x = 180 x = 60; m∠RST = 30° 8 Chapter 1: Line and Angle Relationships 22. 2x x + = 49 3 2 Multiply by LCD, 6 b. ( 90 − (3x − 12) )D = (102 − 3x )D 4 x + 3x = 294 7 x = 294 c. ( 90 − (2 x + 5 y ) )D = (90 − 2 x − 5 y )° x = 42; m∠TSV = x = 21° 2 23. D 29. a. (180 − x ) x + y = 2x − 2 y x + y + 2 x − 2 y = 64 −1x + 3 y = 0 3x − 1 y = 64 −3x + 9 y = 0 3x − y = 64 8 y = 64 y = 8; x = 24 24. D 28. a. ( 90 − x ) 2 x + 3 y = 3x − y + 2 2 x + 3 y + 3x − y + 2 = 80 b. (180 − (3x − 12) )D = (192 − 3x)D c. (180 − (2 x + 5 y ) )D = (180 − 2 x − 5 y )D 30. x − 92 = 92 − 53 x − 92 = 39 x = 131 31. x − 92 + (92 − 53) = 90 x − 92 + 39 = 90 x − 53 = 90 x = 143 32. a. True −1x + 4 y = 2 5 x + 2 y = 78 b. False c. False −5 x + 20 y = 10 5 x + 2 y = 78 22 y = 88 y = 4; x = 14 33. Given: Obtuse ∠MRP JJJG Construct: With OA as one side, an angle ≅ ∠MRP 25. ∠CAB ≅ ∠DAB 26. x + y = 90 x = 12 + y x + y = 90 x − y = 12 = 102 2x x = 51 51 + y = 90 y = 39 ∠s are 51° and 39°. 27. x + y = 180 x = 24 + 2 y 34. Given: Obtuse ∠MRP JJJG Construct: RS , the angle-bisector of ∠MRP x + y = 180 x − 2 y = 24 2 x + 2 y = 360 x − 2 y = 24 = 384 3x x = 128; y = 52 ∠s are 128° and 52°. © 2020 Cengage Learning. All rights reserved. Section 1.2 35. Given: Obtuse ∠MRP Construct: Rays RS, RT, and RU so that ∠MRP is divided into 4 ≅ angles 9 41. m∠1 + m∠2 = 90° If ∠s 1 and 2 are bisected, then 1 2 ⋅ m∠1 + 12 ⋅ m∠2 = 45° 42. Given: Acute ∠1 Construct: ∠2, an angle whose measure is twice that of ∠1 36. Given: Straight angle DEF Construct: a right angle with vertex at E 2 1 1 43. a. 90° b. 90° 37. For the triangle shown, the angle bisectors have been constructed. c. Equal 44. Let m∠USV = x, then m∠TSU = 38 − x 38 − x + 40 = 61 78 − x = 61 78 − 61 = x x = 17; m∠USV = 17° 45. x + 2 z + x − z + 2 x − z = 60 It appears that the angle bisectors meet at one point. HJJJ 38. Given: Acute ∠1 and AB Construct: Triangle ABC which has ∠A ≅ ∠1 , ∠B ≅ ∠1 and side AB 4 x = 60 x = 15 If x = 15, then m ∠ USV = 15 − z , m ∠ VSW = 2(15) − z , and m ∠ USW = 3 x − 6 = 3(15) − 6 = 39 So 15 − z + 2(15) − z = 39 45 − 2 z = 39 6 = 2z z=3 46. a. 52° 39. It appears that the two sides opposite ∠ s A and B are congruent. JJJG 40. Given: Straight ∠ABC and BD Construct: Bisectors of ∠ABD and ∠DBC b. 52° c. Equal 47. 90 + x + x = 360 2 x = 270 x = 135° 48. 90° It appears that a right angle is formed. © 2020 Cengage Learning. All rights reserved. 10 Chapter 1: Line and Angle Relationships SECTION 1.3: Introduction to Geometric Proof 4. 2 x = 12 1. Division Property of Equality or Multiplication Property of Equality 2. Distributive Property [ x + x = (1 + 1) x = 2 x ] 3. Subtraction Property of Equality 5. x = 6 26. 1. x + 3 = 9 5 2. x = 6 5 3. x = 30 4. Addition Property of Equality 27. 1. Given 5. Multiplication Property of Equality 2. Segment-Addition Postulate 6. Addition Property of Equality 3. Subtraction Property of Equality 7. If 2 angles are supplementary, then the sum of their measures is 180°. 28. 1. Given 8. If the sum of the measures of 2 angles is 180°, then the angles are supplementary. 2. The midpoint forms 2 segments of equal measure. 9. Angle-Addition Property 3. Segment-Addition Postulate 10. Definition of angle-bisector 4. Substitution 11. AM + MB = AB 5. Distributive Property AM = MB JJJG 13. EG bisects ∠DEF 12. 14. m∠1 = m∠2 or ∠1 ≅ ∠2 D 15. m∠1 + m∠2 = 90 16. ∠1 and ∠2 are complementary 17. 2 x = 10 18. x=7 19. 7 x + 2 = 30 20. 1 = 50% 2 21. 6 x − 3 = 27 22. x = −20 23. 1. Given 6. Multiplication (or Division) Property of Equality 29. 1. Given 2. If an angle is bisected, then the two angles formed are equal in measure. 3. Angle-Addition Postulate 4. Substitution 5. Distribution Property 6. Multiplication (or Division) Property of Equality 30. 1. Given 2. Angle-Addition Postulate 3. Subtraction Property of Equality 31. S1. M-N-P-Q on MQ R1. Given 2. Distributive Property 2. Segment-Addition Postulate 3. Addition Property of Equality 3. Segment-Addition Postulate 4. Division Property of Equality 4. MN + NP + PQ = MQ 24. 1. Given JJJG JJJG 32. S1. ∠TSW with SU and SV 2. Subtraction Property of Equality R1. Given 3. Division Property of Equality 25. 1. 2( x + 3) − 7 = 11 2. 2 x + 6 − 7 = 11 3. 2 x − 1 = 11 2. Angle-Addition Postulate 3. Angle-Addition Postulate 4. m∠TSW = m∠TSU + m∠USV + m∠VSW © 2020 Cengage Learning. All rights reserved. Section 1.4 11 33. 5 ⋅ x + 5 ⋅ y = 5( x + y ) 34. 5 ⋅ x + 7 ⋅ x = (5 + 7) x = 12 x 35. ( −7)( −2) > 5( −2) or 14 > −10 36. 2. ∠1 ≅ ∠3 4. 1. m∠AOB = m∠1 and m∠BOC = m∠1 2. m∠AOB = m∠BOC 12 < −4 or −3 bc 38. 3. 1. ∠1 ≅ ∠2 and ∠2 ≅ ∠3 x > −5 5. Given: Point N on line s. Construct: Line m through N so that m ⊥ s 39. 1. Given 2. Addition Property of Equality 3. Given 4. Substitution 40. 1. a = b 1. Given 2. a – c = b – c 2. Subtraction Property of Equality 3. c = d 3. Given 4. a – c = b – d 4. Substitution JJJG 6. Given: OA Construct: Right angle BOA (Hint: Use the straightedge to JJJG extend OA to the left.) SECTION 1.4: Relationships: Perpendicular Lines 1. 1. Given 2. If 2 ∠ s are ≅ , then they are equal in measure. 3. Angle-Addition Postulate 4. Addition Property of Equality 7. Given: Line A containing point A Construct: A 45° angle with vertex at A 5. Substitution 6. If 2 ∠ s are = in measure, then they are ≅ . 2. 1. Given 2. The measure of a straight angle is 180°. 3. Angle-Addition Postulate 4. Substitution 5. Given 6. The measure of a right ∠ = 90D . 7. Substitution 8. Subtraction Property of Equality 9. Angle-Addition Postulate 10. Substitution 11. If the sum of measures of 2 angles is 90°, then the angles are complementary. © 2020 Cengage Learning. All rights reserved. 8. Given: AB Construct: The perpendicular bisector of AB 12 Chapter 1: Line and Angle Relationships 9. Given: Triangle ABC Construct: The perpendicular bisectors of sides, AB , AC , and BC 21. a. adjacent b. complementary c. ray AB d. is congruent to e. vertical 22. In space, there is an infinite number of lines perpendicular to a given line at a point on the line. 23. STATEMENTS 1. M -N -P -Q on MQ 2. MN + NQ = MQ 24. AE = AB + BC + CD + DE 25. STATEMENTS REASONS JJJG 1. ∠TSW with SU 1. Given JJJG and SV 2. m∠TSW 2. Angle-Addition = m∠TSU + m∠USW Postulate 3. m∠USW 3. Angle-Addition = m∠USV + m∠VSW Postulate 4. m∠TSW = m∠TSU 4. Substitution + m∠USV + m∠VSW 10. It appears that the perpendicular bisectors meet at one point. 11. R1. Given R3. Substitution S4. m∠1 = m∠2 S5. ∠1 ≅ ∠2 12. R1. Given S2. m∠1 = m∠2 and m∠3 = m∠4 R3. Given S4. m∠2 + m∠3 = 90 REASONS 1. Given 2. Segment-Addition Postulate 3. NP + PQ = NQ 3. Segment-Addition Postulate 4. MN + NP + PQ = MQ 4. Substitution R5. Substitution S6. ∠s 1 and 4 are complementary. 26. m∠GHK = m∠1 + m∠2 + m∠3 + m∠4 14. No; No; Yes 27. In space, there is an infinite number of lines that perpendicularly bisect a given line segment at its midpoint. 15. No; Yes; No 28. 1. Given 13. No; Yes; No 16. No; No; Yes 17. No; Yes; Yes 18. No; No; No 19. a. perpendicular 2. If 2 ∠s are complementary, then the sum of their measures is 90°. 3. Given 4. The measure of an acute angle is between 0 and 90°. b. angles 5. Substitution c. supplementary 6. Subtraction Property of Equality d. right 7. Subtraction Property of Inequality e. measure of angle 8. Addition Property of Inequality 20. a. postulate 9. Transitive Property of Inequality b. union 10. Substitution c. empty set 11. If the measure of an angle is between 0 and 90°, then the angle is an acute ∠. d. less than e. point © 2020 Cengage Learning. All rights reserved. Section 1.5 13 29. Angles 1, 2, 3, and 4 are adjacent and form the straight angle AOB, which measures 180. Therefore, m∠1 + m∠2 + m∠3 + m∠4 = 180. HJJG HJJG 13. Given: AB ⊥ CD Prove: ∠AEC is a right angle. 30. If ∠2 and ∠3 are complementary, then m∠2 + m∠3 = 90. From Exercise 29, m∠1 + m∠2 + m∠3 + m∠4 = 180. Therefore, m∠1 + m∠4 = 90 and ∠1 and ∠4 are complementary. SECTION 1.5: The Formal Proof of a Theorem Figure for exercises 13 and 14. 1. H: A line segment is bisected. C: Each of the equal segments has half the length of the original segment. 2. H: Two sides of a triangle are congruent. C: The triangle is isosceles. 3. First write the statement in the “If, then” form. If a figure is a square, then it is a quadrilateral. 14. Given: ∠AEC is a right angle HJJG HJJG Prove: AB ⊥ CD 15. Given: ∠1 is complementry to ∠3 ∠2 is complementry to ∠3 Prove: ∠1 ≅ ∠2 H: A figure is a square. C: It is a quadrilateral. 4. First write the statement in the “If, then” form. If a polygon is a regular polygon, then it has congruent interior angles. 16. Given: ∠1 is supplementary to ∠3 ∠2 is supplementary to ∠3 Prove: ∠1 ≅ ∠2 H: A polygon is a regular polygon. C: It has congruent interior angles. 5. First write the statement in the “If, then” form. If each is right angle, then two angles are congruent. H: Each is a right angle. C: Two angles are congruent. 17. Given: Lines l and m intersect as shown Prove: ∠1 ≅ ∠2 and ∠3 ≅ ∠4 6. First write the statement in the “If, then” form. If polygons are similar, then the lengths of corresponding sides are proportional. H: Polygons are similar. C: The lengths of corresponding sides are proportional. 18. Given: ∠1 and ∠2 are right angles Prove: ∠1 ≅ ∠2 7. Statement, Drawing, Given, Prove, Proof 8. a. Hypothesis b. Hypothesis c. Conclusion 9. a. Given 19. m∠2 = 55D , m∠3 = 125D , m∠4 = 55D b. Prove 10. a, c, d 11. After the theorem has been proved. 12. No © 2020 Cengage Learning. All rights reserved. 20. m∠1 = 133D , m∠3 = 133D , m∠4 = 47D 21. m∠1 = m∠3 3x + 10 = 4 x − 30 x = 40; m∠1 = 130D 14 22. Chapter 1: Line and Angle Relationships m∠2 = m∠4 6x + 8 = 7 x x = 8; m∠2 = 56° 28. Given: ∠1 is supplementary to ∠2 ∠3 is supplementary to ∠2 Prove: ∠1 ≅ ∠3 23. m∠1 + m∠2 = 180° 2 x + x = 180 3 x = 180 x = 60; m∠1 = 120 24. m∠2 + m∠3 = 180D x + 15 + 2 x = 180 3 x = 165 x = 55; m∠2 = 70° 25. x x − 10 + + 40 = 180 2 3 x x + + 30 = 180 2 3 x x + = 150 2 3 Multiply by 6 3x + 2 x = 900 5 x = 900 x = 180; m∠2 = 80° 26. x = 180 3 x x + = 160 3 x + 20 + STATEMENTS REASONS 1. ∠1 is supplementary to ∠2 1. Given ∠3 is supplementary to ∠2 2. m∠1+ m∠2 = 180 2. If 2 ∠s are supplementary, m∠3 + m∠2 = 180 then the sum of their measures is 180. 3. m∠1+ m∠2 3. Substitution = m∠3 + m∠2 4. m∠1 = m∠3 4. Subtraction Property of Equality 5. ∠1 ≅ ∠3 5. If 2 ∠s are = in measure, then they are ≅ . 29. If 2 lines intersect, the vertical angles formed are congruent. HJJG HJJG Given: AB and CD intersect at E Prove: ∠1 ≅ ∠2 Multiply by 3 3x + x = 480 4 x = 480 x = 120; m∠4 = 40° 27. 1. Given 2. If 2 ∠ s are complementary, the sum of their measures is 90. 3. Substitution STATEMENTS REASONS HJJG HJJG 1. AB and CD 1. Given intersect at E 2. ∠1 is supplementary to ∠AED 2. If the exterior sides ∠2 is supplementary to ∠AED of two adjacent ∠s form a straight line, then these ∠s are supplementary 3. ∠1 ≅ ∠2 3. If 2 ∠s are supplementary to the same ∠, then these ∠s are ≅ . 4. Subtraction Property of Equality 5. If 2 ∠ s are = in measure, then they are ≅ . © 2020 Cengage Learning. All rights reserved. Section 1.5 30. Any two right angles are congruent. Given: ∠1 is a right ∠ ∠2 is a right ∠ Prove: ∠1 ≅ ∠2 15 32. If 2 segments are congruent, then their midpoints separate these segments into four congruent segments. Given: AB ≅ DC M is the midpoint of AB N is the midpoint of DC Prove: AM ≅ MB ≅ DN ≅ NC STATEMENTS REASONS 1. ∠1 is a right ∠ 1. Given ∠2 is a right ∠ 2. m∠1 = 90 2. Measure of a right m∠2 = 90 ∠ = 90. 3. m∠1 = m∠2 3. Substitution 4. ∠1 ≅ ∠2 4. If 2 ∠s are = in measure, then they are ≅ . 31. R1. Given S2. ∠ABC is a right ∠ . R3. The measure of a right ∠ = 90 . R4. Angle-Addition Postulate STATEMENTS 1. AB ≅ DC 2. AB = DC 3. AB = AM + MB DC = DN + NC 4. AM + MB = DN + NC 1. Given 2. If 2 segments are ≅ , then their lengths are = . 3. Segment-Addition Postulate 4. Substitution 5. M is the midpoint of AB 5. Given N is the midpoint of DC 6. AM = MB and DN = NC S6. ∠1 is complementary to ∠2 . 7. AM + AM = DN + DN or 2 ⋅ AM = 2 ⋅ DN 8. AM = DN 9. AM = MB = DN = NC 10. AM ≅ MB ≅ DN ≅ NC © 2020 Cengage Learning. All rights reserved. REASONS 6. If a point is the midpoint of a segment, it forms 2 segments equal in measure. 7. Substitution 8. Division Property of Equality 9. Substitution 10. If segments are = in length, then they are ≅ . 16 Chapter 1: Line and Angle Relationships 33. If 2 angles are congruent, then their bisectors separate these angles into four congruent angles. Given: ∠ABC ≅ ∠EFG JJJG BD bisects ∠ABC JJJG FH bisects ∠EFG Prove: ∠1 ≅ ∠2 ≅ ∠3 ≅ ∠4 STATEMENTS 1. ∠ABC ≅ ∠EFG 2. m∠ABC = m∠EFG REASONS 1. Given 2. If 2 angles are ≅ , their measures are = . 3. m∠ABC = m∠1+ m∠2 3. Angle-Addition m∠EFG = m∠3 + m∠4 Postulate 4. m∠1+ m∠2 4. Substitution = m∠3 + m∠4 JJJG 5. Given 5. BD bisects ∠ABC JJJG FH bisects ∠EFG 6. m∠1= m∠2 and 6. If a ray bisects m∠3 = m∠4 an ∠, then 2 ∠s of equal measure are formed. 7. m∠1+ m∠1 7. Substitution = m∠3 + m∠3 or 2 ⋅ m∠1 = 2 ⋅ m∠3 8. m∠1= m∠3 8. Division Property of Equality 9. m∠1= m∠2 9. Substitution = m∠3 = m∠4 10. ∠1 ≅ ∠2 ≅ ∠3 ≅ ∠4 10. If ∠s are = in measure, then they are ≅ . 34. The bisectors of two adjacent supplementary angles form a right angle. Given: ∠ABC is supplementary to ∠CBD JJJG BE bisects ∠ABC JJJG BF bisects ∠CBD Prove: ∠EBF is a right angle STATEMENTS 1. ∠ABC is supplementary to ∠CBD 2. m∠ABC + m∠CBD =180 3. m∠ABC = m∠1+ m∠2 m∠CBD = m∠3 + m∠4 4. m∠1+ m∠2 + m∠3 + m∠4 =180 JJJG 5. BE bisects ∠ABC JJJG BF bisects ∠CBD 6. m∠1= m∠2 and m∠3 = m∠4 7. m∠2 + m∠2 + m∠3 + m∠3 =180 or 2 ⋅ m∠2 + 2 ⋅ m∠3 =180 8. m∠2 + m∠3 = 90 9. m∠EBF = m∠2 + m∠3 10. m∠EBF = 90 11. ∠EBF is a right angle REASONS 1. Given 2. The sum of the measures of supplementary angles is 180. 3. Angle-Addition Postulate 4. Substitution 5. Given 6. If a ray bisects an ∠, then 2 ∠s of equal measure are formed. 7. Substitution 8. Division Property of Equality 9. Angle-Addition Postulate 10. Substitution 11. If the measure of an ∠ is 90, then the ∠ is a right ∠. © 2020 Cengage Learning. All rights reserved.