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

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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 โข Brazil โข Mexico โข Singapore โข United Kingdom โข United States
ยฉ 2020 Cengage Learning
ISBN: 978-0-357-02221-4
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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
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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 studentsโ 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 1โ8. 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 1โ8 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 1โ11 as well as topics desired from Appendix A and/or
The Introduction to Logic (see website).
Daniel C. Alexander
Geralyn M. Koeberlein
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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 โIf . . ., then . . .โ 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 segmentโs 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 โbranch of mathematics,โ 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
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utilized throughout the textbook. The instructor should alert students to undefined terms
such as โbuilding blocks.โ 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 algebraโs 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 โperpendicular relationshipโ 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 propertiesโโparticularly 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
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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 โthe Earth is flat,โ 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, โIf two parallel lines are cut by
a transversal, then . . . .โ
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, โIf . . . , then these lines are parallel.โ 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, โThe sum of the measures of the interior angles of a triangle is
180ยฐ.โ 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 โ 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
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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 studentโs 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 โis
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.โ 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, โCorresponding
Parts of Congruent Triangles are Congruent.โ 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
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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 45ยฐ or 60ยฐ).
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, โIf a quadrilateral is a parallelogram, then . . . .โ 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, โIf a
quadrilateral . . . , then the quadrilateral is a parallelogram.โ 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
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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 โCorresponding Angles of Similar Triangles are Congruent,โ while
(2.) CSSTP means โCorresponding Sides of Similar Triangles are Proportional.โ
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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 45ยฐ, 45ยฐ, and 90ยฐ always
has congruent legs while the hypotenuse is 2 times as long as either leg. Also, a right
triangle with angle measures of 30ยฐ, 60ยฐ, and 90ยฐ 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, Cevaโs 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
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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 โseeโ 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
6.3.5โ6.3.7 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: โlocation.โ 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.
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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 โstepping stoneโ 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.
Heronโs 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 Heronโs 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 ฯ as a ratio and then provide some approximations of its
22
C
and 3.1416). Using ฯ = , we can show that C = ฯ d and C = 2 ฯ 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 = ฯ r 2 . Note that the concept of limit needs a few examples. For
students to distinguish between 2 ฯ r and ฯ r 2 (for circumference and area), have them
compare units, where r = 3 cm, 2 ฯ r = 2 ฯ โ
3 cm = 6 ฯ cm (a linear measure) while ฯ r 2
= ฯ โ
3 cm โ
3 cm or 9 ฯ 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 60ยฐ, 90ยฐ, or 120ยฐ; otherwise,
xiii
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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 = ฯ r 2 h); likewise compare the volume formulas
1
1
1
for the pyramid (V = Bh) and right circular cylinder (V = Bh or V = ฯ 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 Eulerโs 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
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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 โwarm upโ 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, โIf a quadrilateral is a
parallelogram, then its diagonals bisect each other.โ 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(โa,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
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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 60ยฐ =
, and sin 90ยฐ = 1.
memory: sin 0ยฐ = 0, sin 30ยฐ = , sin 45ยฐ =
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 0ยฐ = 1, cos 30ยฐ =
, 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 ฮธ + cos 2 ฮธ = 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
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Section 11.3: The Tangent Ratio and Other Ratios
opposite
Students should memorize the tangent ratio as
and memorize exact values for
adjacent
tan 0ยฐ, tan 30ยฐ, tan 45ยฐ, etc. Some attention and discussion should be devoted to the claim
that โtan 90ยฐ is undefined.โ 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 ฮธ = , then csc ฮธ = .
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 34ยฐ (as reciprocal of cos 34ยฐ). 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 ฮฑ , 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
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Chapter 1 Line and Angle Relationships
SECTION 1.1: Early Definitions and
Postulates
1. AC
2. Midpoint
3. 6.25 ft โ
12 in./ft = 75 in.
4. 52 in. รท 12 in./ft = 4 1 ft or 4 ft 4 in.
3
5.
1 m โ
3.28 ft/m = 1.64 feet
2
6. 16.4 ft รท 3.28 ft/m = 5 m
7. 18 โ 15 = 3 mi
8. 300 + 450 + 600 = 1350 ft
1350 ft รท 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 โ 2
โ x = โ3
x=3
AM = 7
16. 2( x + 1) = 3( x โ 2)
2 x + 2 = 3x โ 6
โ 1x = โ 8
x =8
AB = AM + MB
AB = 18 + 18 = 36
17. 2 x + 1 + 3x + 2 = 6 x โ 4
5x + 3 = 6x โ 4
โ1x = โ7
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
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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 โ 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
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Section 1.2
34. A segment can be divided into 2n congruent
parts, where n โฅ 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 โ FAE ; the AngleAddition Postulate does not apply.
13. mโ FAC + mโ CAD = 180
โ FAC and โ 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. 62ยฐ
17. 42ยฐ
18. 2 x + 9 + 3x โ 2 = 67
5 x + 7 = 67
5 x = 60
x = 12
19. 2 x โ 10 + x + 6 = 4( x โ 6)
3 x โ 4 = 4 x โ 24
20 = x
x = 20
mโ RSV = 4(20 โ 6) = 56ยฐ
20. 5( x + 1) โ 3 + 4( x โ 2) + 3 = 4(2 x + 3) โ 7
5 x + 5 โ 3 + 4 x โ 8 + 3 = 8 x + 12 โ 7
9 x โ 3 = 8x + 5
x =8
mโ RSV = 4(2 โ
8 + 3) โ 7 = 69ยฐ
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
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2 x + x = 180
3 x = 180
x = 60; mโ RST = 30ยฐ
8
Chapter 1: Line and Angle Relationships
22.
2x x
+ = 49
3 2
Multiply by LCD, 6
b.
( 90 โ (3x โ 12) )D = (102 โ 3x )D
4 x + 3x = 294
7 x = 294
c.
( 90 โ (2 x + 5 y ) )D = (90 โ 2 x โ 5 y )ยฐ
x = 42; mโ TSV = x = 21ยฐ
2
23.
D
29. a. (180 โ x )
x + y = 2x โ 2 y
x + y + 2 x โ 2 y = 64
โ1x + 3 y = 0
3x โ 1 y = 64
โ3x + 9 y = 0
3x โ y = 64
8 y = 64
y = 8; x = 24
24.
D
28. a. ( 90 โ x )
2 x + 3 y = 3x โ y + 2
2 x + 3 y + 3x โ y + 2 = 80
b.
(180 โ (3x โ 12) )D = (192 โ 3x)D
c.
(180 โ (2 x + 5 y ) )D = (180 โ 2 x โ 5 y )D
30.
x โ 92 = 92 โ 53
x โ 92 = 39
x = 131
31.
x โ 92 + (92 โ 53) = 90
x โ 92 + 39 = 90
x โ 53 = 90
x = 143
32. a. True
โ1x + 4 y = 2
5 x + 2 y = 78
b. False
c. False
โ5 x + 20 y = 10
5 x + 2 y = 78
22 y = 88
y = 4; x = 14
33. Given: Obtuse โ MRP
JJJG
Construct: With OA as one side,
an angle โ
โ MRP
25. โ CAB โ
โ DAB
26.
x + y = 90
x = 12 + y
x + y = 90
x โ y = 12
= 102
2x
x = 51
51 + y = 90
y = 39
โ s are 51ยฐ and 39ยฐ.
27.
x + y = 180
x = 24 + 2 y
34. Given: Obtuse โ MRP
JJJG
Construct: RS , the angle-bisector of โ MRP
x + y = 180
x โ 2 y = 24
2 x + 2 y = 360
x โ 2 y = 24
= 384
3x
x = 128; y = 52
โ s are 128ยฐ and 52ยฐ.
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Section 1.2
35. Given: Obtuse โ MRP
Construct: Rays RS, RT, and RU so that โ MRP
is divided into 4 โ
angles
9
41. mโ 1 + mโ 2 = 90ยฐ
If โ s 1 and 2 are bisected, then
1
2
โ
mโ 1 + 12 โ
mโ 2 = 45ยฐ
42. Given: Acute โ 1
Construct: โ 2, an angle whose measure is twice
that of โ 1
36. Given: Straight angle DEF
Construct: a right angle with vertex at E
2
1
1
43. a. 90ยฐ
b. 90ยฐ
37. For the triangle shown, the angle bisectors have
been constructed.
c. Equal
44. Let mโ USV = x, then mโ TSU = 38 โ x
38 โ x + 40 = 61
78 โ x = 61
78 โ 61 = x
x = 17; mโ USV = 17ยฐ
45. x + 2 z + x โ z + 2 x โ z = 60
It appears that the angle bisectors meet at one
point.
HJJJ
38. Given: Acute โ 1 and AB
Construct: Triangle ABC which has
โ A โ
โ 1 , โ B โ
โ 1 and side AB
4 x = 60
x = 15
If x = 15, then m โ USV = 15 โ z ,
m โ VSW = 2(15) โ z , and
m โ USW = 3 x โ 6 = 3(15) โ 6 = 39
So 15 โ z + 2(15) โ z = 39
45 โ 2 z = 39
6 = 2z
z=3
46. a. 52ยฐ
39. It appears that the two sides opposite โ s A and B
are congruent.
JJJG
40. Given: Straight โ ABC and BD
Construct: Bisectors of โ ABD and โ DBC
b. 52ยฐ
c. Equal
47. 90 + x + x = 360
2 x = 270
x = 135ยฐ
48. 90ยฐ
It appears that a right angle is formed.
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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 180ยฐ.
28. 1. Given
8. If the sum of the measures of 2 angles is 180ยฐ,
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 โ DEF
12.
14. mโ 1 = mโ 2 or โ 1 โ
โ 2
D
15. mโ 1 + mโ 2 = 90
16. โ 1 and โ 2 are complementary
17. 2 x = 10
18.
x=7
19. 7 x + 2 = 30
20.
1 = 50%
2
21. 6 x โ 3 = 27
22.
x = โ20
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. โ TSW with SU and SV
2. Subtraction Property of Equality
R1. Given
3. Division Property of Equality
25. 1. 2( x + 3) โ 7 = 11
2. 2 x + 6 โ 7 = 11
3. 2 x โ 1 = 11
2. Angle-Addition Postulate
3. Angle-Addition Postulate
4. mโ TSW = mโ TSU + mโ USV + mโ VSW
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Section 1.4
11
33. 5 โ
x + 5 โ
y = 5( x + y )
34. 5 โ
x + 7 โ
x = (5 + 7) x = 12 x
35. ( โ7)( โ2) > 5( โ2) or 14 > โ10
36.
2. โ 1 โ
โ 3
4. 1. mโ AOB = mโ 1 and mโ BOC = mโ 1
2. mโ AOB = mโ BOC
12 < โ4 or โ3 bc
38.
3. 1. โ 1 โ
โ 2 and โ 2 โ
โ 3
x > โ5
5. Given: Point N on line s.
Construct: Line m through N so that m โฅ s
39. 1. Given
2. Addition Property of Equality
3. Given
4. Substitution
40. 1. a = b
1. Given
2. a โ c = b โ c
2. Subtraction Property of
Equality
3. c = d
3. Given
4. a โ c = b โ 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 โ s are โ
, then they are equal in
measure.
3. Angle-Addition Postulate
4. Addition Property of Equality
7. Given: Line A containing point A
Construct: A 45ยฐ angle with vertex at A
5. Substitution
6. If 2 โ s are = in measure, then they are โ
.
2. 1. Given
2. The measure of a straight angle is 180ยฐ.
3. Angle-Addition Postulate
4. Substitution
5. Given
6. The measure of a right โ = 90D .
7. Substitution
8. Subtraction Property of Equality
9. Angle-Addition Postulate
10. Substitution
11. If the sum of measures of 2 angles is 90ยฐ, then
the angles are complementary.
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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. โ TSW with SU
1. Given
JJJG
and SV
2. mโ TSW
2. Angle-Addition
= mโ TSU + mโ USW
Postulate
3. mโ USW
3. Angle-Addition
= mโ USV + mโ VSW
Postulate
4. mโ TSW = mโ TSU
4. Substitution
+ mโ USV + mโ VSW
10. It appears that the perpendicular bisectors meet at
one point.
11. R1. Given
R3. Substitution
S4. mโ 1 = mโ 2
S5. โ 1 โ
โ 2
12. R1. Given
S2. mโ 1 = mโ 2 and mโ 3 = mโ 4
R3. Given
S4. mโ 2 + mโ 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. โ s 1 and 4 are complementary.
26. mโ GHK = mโ 1 + mโ 2 + mโ 3 + mโ 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 โ s are complementary, then the sum of
their measures is 90ยฐ.
3. Given
4. The measure of an acute angle is between 0
and 90ยฐ.
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
90ยฐ, then the angle is an acute โ .
d. less than
e. point
ยฉ 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, mโ 1 + mโ 2 + mโ 3 + mโ 4 = 180.
HJJG HJJG
13. Given: AB โฅ CD
Prove: โ AEC is a right angle.
30. If โ 2 and โ 3 are complementary, then
mโ 2 + mโ 3 = 90. From Exercise 29,
mโ 1 + mโ 2 + mโ 3 + mโ 4 = 180. Therefore,
mโ 1 + mโ 4 = 90 and โ 1 and โ 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 โIf, thenโ form.
If a figure is a square, then it is a quadrilateral.
14. Given: โ AEC is a right angle
HJJG HJJG
Prove: AB โฅ CD
15. Given: โ 1 is complementry to โ 3
โ 2 is complementry to โ 3
Prove: โ 1 โ
โ 2
H: A figure is a square.
C: It is a quadrilateral.
4. First write the statement in the โIf, thenโ form.
If a polygon is a regular polygon, then it has
congruent interior angles.
16. Given: โ 1 is supplementary to โ 3
โ 2 is supplementary to โ 3
Prove: โ 1 โ
โ 2
H: A polygon is a regular polygon.
C: It has congruent interior angles.
5. First write the statement in the โIf, thenโ 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: โ 1 โ
โ 2 and โ 3 โ
โ 4
6. First write the statement in the โIf, thenโ 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: โ 1 and โ 2 are right angles
Prove: โ 1 โ
โ 2
7. Statement, Drawing, Given, Prove, Proof
8. a. Hypothesis
b. Hypothesis
c. Conclusion
9. a. Given
19. mโ 2 = 55D , mโ 3 = 125D , mโ 4 = 55D
b. Prove
10. a, c, d
11. After the theorem has been proved.
12. No
ยฉ 2020 Cengage Learning. All rights reserved.
20. mโ 1 = 133D , mโ 3 = 133D , mโ 4 = 47D
21.
mโ 1 = mโ 3
3x + 10 = 4 x โ 30
x = 40; mโ 1 = 130D
14
22.
Chapter 1: Line and Angle Relationships
mโ 2 = mโ 4
6x + 8 = 7 x
x = 8; mโ 2 = 56ยฐ
28. Given: โ 1 is supplementary to โ 2
โ 3 is supplementary to โ 2
Prove: โ 1 โ
โ 3
23. mโ 1 + mโ 2 = 180ยฐ
2 x + x = 180
3 x = 180
x = 60; mโ 1 = 120
24. mโ 2 + mโ 3 = 180D
x + 15 + 2 x = 180
3 x = 165
x = 55; mโ 2 = 70ยฐ
25.
x
x
โ 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; mโ 2 = 80ยฐ
26.
x
= 180
3
x
x + = 160
3
x + 20 +
STATEMENTS
REASONS
1. โ 1 is supplementary to โ 2 1. Given
โ 3 is supplementary to โ 2
2. mโ 1+ mโ 2 = 180
2. If 2 โ s are supplementary,
mโ 3 + mโ 2 = 180
then the sum of their
measures is 180.
3. mโ 1+ mโ 2
3. Substitution
= mโ 3 + mโ 2
4. mโ 1 = mโ 3
4. Subtraction Property
of Equality
5. โ 1 โ
โ 3
5. If 2 โ s are = in
measure, then they
are โ
.
29. If 2 lines intersect, the vertical angles formed are
congruent.
HJJG
HJJG
Given: AB and CD intersect at E
Prove: โ 1 โ
โ 2
Multiply by 3
3x + x = 480
4 x = 480
x = 120; mโ 4 = 40ยฐ
27. 1. Given
2. If 2 โ 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. โ 1 is supplementary to โ AED 2. If the exterior sides
โ 2 is supplementary to โ AED
of two adjacent โ s form
a straight line, then
these โ s are supplementary
3. โ 1 โ
โ 2
3. If 2 โ s are supplementary to
the same โ , then
these โ s are โ
.
4. Subtraction Property of Equality
5. If 2 โ s are = in measure, then they are โ
.
ยฉ 2020 Cengage Learning. All rights reserved.
Section 1.5
30. Any two right angles are congruent.
Given: โ 1 is a right โ
โ 2 is a right โ
Prove: โ 1 โ
โ 2
15
32. If 2 segments are congruent, then their midpoints
separate these segments into four congruent
segments.
Given: AB โ
DC
M is the midpoint of AB
N is the midpoint of DC
Prove: AM โ
MB โ
DN โ
NC
STATEMENTS
REASONS
1. โ 1 is a right โ 1. Given
โ 2 is a right โ
2. mโ 1 = 90
2. Measure of a right
mโ 2 = 90
โ = 90.
3. mโ 1 = mโ 2
3. Substitution
4. โ 1 โ
โ 2
4. If 2 โ s are = in
measure, then they
are โ
.
31. R1. Given
S2. โ ABC is a right โ .
R3. The measure of a right โ = 90 .
R4. Angle-Addition Postulate
STATEMENTS
1. AB โ
DC
2. AB = DC
3. AB = AM + MB
DC = DN + NC
4. AM + MB = DN + NC
1. Given
2. If 2 segments are
โ
, 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. โ 1 is complementary to โ 2 .
7. AM + AM = DN + DN
or 2 โ
AM = 2 โ
DN
8. AM = DN
9. AM = MB = DN = NC
10. AM โ
MB โ
DN โ
NC
ยฉ 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 โ
.
16
Chapter 1: Line and Angle Relationships
33. If 2 angles are congruent, then their bisectors
separate these angles into four congruent angles.
Given: โ ABC โ
โ EFG
JJJG
BD bisects โ ABC
JJJG
FH bisects โ EFG
Prove: โ 1 โ
โ 2 โ
โ 3 โ
โ 4
STATEMENTS
1. โ ABC โ
โ EFG
2. mโ ABC = mโ EFG
REASONS
1. Given
2. If 2 angles are
โ
, their
measures are = .
3. mโ ABC = mโ 1+ mโ 2 3. Angle-Addition
mโ EFG = mโ 3 + mโ 4
Postulate
4. mโ 1+ mโ 2
4. Substitution
= mโ 3 + mโ 4
JJJG
5. Given
5. BD bisects โ ABC
JJJG
FH bisects โ EFG
6. mโ 1= mโ 2 and
6. If a ray bisects
mโ 3 = mโ 4
an โ , then 2 โ s
of equal measure
are formed.
7. mโ 1+ mโ 1
7. Substitution
= mโ 3 + mโ 3 or
2 โ
mโ 1 = 2 โ
mโ 3
8. mโ 1= mโ 3
8. Division Property
of Equality
9. mโ 1= mโ 2
9. Substitution
= mโ 3 = mโ 4
10. โ 1 โ
โ 2 โ
โ 3 โ
โ 4
10. If โ s are = in
measure, then
they are โ
.
34. The bisectors of two adjacent supplementary
angles form a right angle.
Given: โ ABC is supplementary to โ CBD
JJJG
BE bisects โ ABC
JJJG
BF bisects โ CBD
Prove: โ EBF is a right angle
STATEMENTS
1. โ ABC is supplementary
to โ CBD
2. mโ ABC + mโ CBD
=180
3. mโ ABC = mโ 1+ mโ 2
mโ CBD = mโ 3 + mโ 4
4. mโ 1+ mโ 2 + mโ 3
+ mโ 4 =180
JJJG
5. BE bisects โ ABC
JJJG
BF bisects โ CBD
6. mโ 1= mโ 2 and
mโ 3 = mโ 4
7. mโ 2 + mโ 2 + mโ 3
+ mโ 3 =180 or
2 โ
mโ 2 + 2 โ
mโ 3 =180
8. mโ 2 + mโ 3 = 90
9. mโ EBF = mโ 2 + mโ 3
10. mโ EBF = 90
11. โ 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 โ , then 2 โ 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 โ is 90, then
the โ is a right โ .
ยฉ 2020 Cengage Learning. All rights reserved.

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