Answers
Geometry Copyright © Big Ideas Learning, LLC Answers All rights
reserved. A22
127. ( )( )2 7x x+ − 128. ( )( )12 11x x− +
129. ( )( )7 4x x− + 130. ( )( )5 3x x− −
131. ( )( )2 3x x− − 132. ( )( )4 9x x+ −
133. ( )( )1 1x x+ − 134. ( )( )3 3x x+ −
135. ( )( )5 5x x+ − 136. ( )( )2 3 4x x− +
137. ( )( )3 5 7x x− − 138. ( )( )5 2 4x x+ +
139. 8 and 3x x= − = −
140. 6 and 2x x= − =
141. 12 and 1x x= − =
142. 9 and 8x x= − = −
143. 5 and 4x x= − =
144. 10 and 7x x= − = −
145. 3 and 1x x= − =
146. 5 and 2x x= − =
147. 11 and 1x x= − =
148. 32 and 5x x= − =
149. 45 and 3x x= − =
150. 713 2 and x x= − =
151. a. 7 words per min b. 87.5 words c. 105 words d. 17.5
words
Chapter 3 3.1 Start Thinking
right triangle; no; no; Because points B and C connect
perpendicular lines, you cannot plot either point to make a
perpendicular segment or a parallel segment.
3.1 Warm Up
1. Sample answer: BC
2. GE
3. CG
4. , ,AB BC BD
5. Sample answer: andFE FG
6. Sample answer: D
3.1 Cumulative Review Warm Up 1. ( )4, 11K 2. ( )27, 18J − − 3.
( )21, 2K −
3.1 Practice A
1. andAB CD
2. andAC CD
3. no; AB CD
and by the Parallel Postulate (Post. 3.1), there is exactly one
line parallel to AB
through
point C.
4. no; They are intersecting lines.
5. 2∠ and 8,∠ 3∠ and 5∠
6. 1∠ and 7,∠ 4∠ and 6∠
7. 1∠ and 5,∠ 2∠ and 6,∠ 3∠ and 7,∠ 4∠ and 8∠
8. 2∠ and 5,∠ 3∠ and 8∠
9. no; By definition, skew lines are not coplaner.
10. 2 pairs; 4 pairs; ( )2 2n − pairs
11. a. AB
and ,CD
AC
and BD
b. AC
and ,CD
BD
and CD
c. 2∠ and 5,∠ 3∠ and 8∠
CA
B
Answers
Copyright © Big Ideas Learning, LLC Geometry All rights
reserved. Answers
A23
d. 1∠ and 5,∠ 2∠ and 6,∠ 3∠ and 7,∠ 4∠ and 8∠
e. 2∠ and 8,∠ 3∠ and 5∠ f. 1∠ and 7,∠ 4∠ and 6∠
3.1 Practice B 1. lines c and d 2. lines e and f
3. Sample answer: lines c and e
4. planes A and B
5. no; lines f and g appear to be coplanar and although they do
not intersect, there is not enough information to determine that
the lines are parallel.
6. no; lines e and g appear to be coplanar and intersect at a
90° angle, but there is not enough information to determine that
the lines are perpendicular.
7. alternate interior 8. corresponding
9. alternate exterior 10. corresponding
11. consecutive exterior
12. no; The lines do not intersect, however they could be
coplanar to a third plane.
13. a. true; The road and the sidewalk appear to lie in the same
plane and they do not intersect. b. false; The road and the
crosswalk appear to
intersect. c. true; A properly installed stop sign intersects
the
ground at a 90° angle.
3.1 Enrichment and Extension 1. yes; The two lines of
intersection are coplanar
because they are both in the third plane. The two lines do not
intersect because they are in parallel planes. Because they are
coplanar and do not intersect, they are parallel.
2.
Line a appears to be parallel to line c; If two lines are
parallel to the same line, then they are parallel to each
other.
3.
Line seems to be parallel to line n; If two lines are
perpendicular to the same line, then they are parallel to each
other.
4.
5.
6. a. 5, 11, 17∠ ∠ ∠
b. 5, 9, 17∠ ∠ ∠
c. 8, 12, 17∠ ∠ ∠
d. 7, 9, 18∠ ∠ ∠
e. 2, 10, 14∠ ∠ ∠
f. 4, 10, 16∠ ∠ ∠ g. 3, 11, 15∠ ∠ ∠
h. 15∠
3.1 Puzzle Time A YARDSTICK
m
n
m
n
P
B
A
a
b
c
Answers
Geometry Copyright © Big Ideas Learning, LLC Answers All rights
reserved. A24
3.2 Start Thinking
one angle measure; With the measurement of one of the angles,
you can use the properties of corresponding angles, alternative
interior angles, alternate exterior angles, and consecutive
interior angles to find the other seven measurements.
3.2 Warm Up 1. 34° 2. 17° 3. 147°
4. 53° 5. 86° 6. 84°
3.2 Cumulative Review Warm Up 1.
2.
3.
4.
3.2 Practice A 1. 1 87 , 2 93 ; 1 87m m m∠ = ° ∠ = ° ∠ = ° by
the
Alternate Interior Angles Theorem (Thm. 3.2). 2 93m∠ = ° by the
Consecutive Interior Angles
Theorem (Thm. 3.4).
2. 1 78 , 2 78 ; 1 78m m m∠ = ° ∠ = ° ∠ = ° by the Corresponding
Angles Theorem (Thm. 3.1).
2 78m∠ = ° by the Alternate Exterior Angles Theorem (Thm.
3.3).
3. ( )8; 37 6 1148 68
x
xx
° = − °
==
4. 10;
( )2 142 180
2 9 142 180
2 18 142 1802 160 180
2 2010
m
x
xx
xx
∠ + ° = °
+ ° + ° = °
+ + =+ =
==
5. 1 112 , 2 68 , 3 112 ;m m m∠ = ° ∠ = ° ∠ = ° Because the 112°
angle is a vertical angle to 1,∠ by the Vertical Angles Congruence
Theorem (Thm. 2.6) they are congruent. Because 1∠ and 2∠ are
consecutive interior angles, they are supplementary by the
Consecutive Interior Angles Theorem (Thm. 3.4). Because the given
112° angle and 3∠are alternate exterior angles, they are congruent
by the Alternate Exterior Angles Theorem (Thm. 3.3).
6. 1 45 , 2 45 , 3 135 ;m m m∠ = ° ∠ = ° ∠ = ° Because the given
45° angle is a corresponding angle with
1,∠ and 1∠ is a corresponding angle with 2∠ , they are all
congruent by the Corresponding Angles Theorem (Thm. 3.1). Because
the 45° angle is a consecutive interior angle with 3,∠ they are
supplementary by the Consecutive Angles Theorem (Thm. 3.4).
1 2
3 4
m
t
5 6
7 8
45°
R
R
A
B
CD
R
A B
C
D
R S
x y
Answers
Copyright © Big Ideas Learning, LLC Geometry All rights
reserved. Answers
A25
7.
8.
1 90 ;m∠ = ° Because 1∠ is congruent and supplementary to 2,∠
the measure of each angle is 90°.
3.2 Practice B 1. 1 41 , 2 41 ; 1 41m m m∠ = ° ∠ = ° ∠ = ° by
the
Corresponding Angles Theorem (Thm. 3.1). 2 41m∠ = ° by the
Vertical Angles Congruence
Theorem (Thm. 2.6).
2. 1 124 , 2 124 ; 1 124m m m∠ = ° ∠ = ° ∠ = ° by the Alternate
Exterior Angles Theorem (Thm. 3.3).
2 124m∠ = ° by the Vertical Angles Congruence Theorem (Thm.
2.6).
3. 16; ( ) ( )24 3 832 332 216
x x
x xx
x
+ ° = − °
+ ===
4. 51; ( ) ( )2 27 3 25 1803
2 18 3 25 1803
11 7 1803
51
x x
x x
x
x
+ ° + − ° = °
+ + − =
− =
=
5. 1 102 , 2 102 , 3 78 ;m m m∠ = ° ∠ = ° ∠ = ° Because the
given 102° angle is an alternate interior angle with 1,∠ they are
congruent by the Alternate Interior Angles Congruence Theorem (Thm.
3.2). Because the given 102° angle and 2∠ are alternate exterior
angles, they are congruent by the Alternate Exterior Angles Theorem
(Thm. 3.3). Because 2∠and 3∠ are a linear pair, they are
supplementary by the Linear Pair Postulate (Post. 2.8).
6. 1 68 , 2 68 , 3 112 ;m m m∠ = ° ∠ = ° ∠ = ° Because the given
68° angle and 1∠ are corresponding angles, they are congruent by
the Corresponding Angles Theorem (Thm. 3.1). Because 1∠ and 2∠ are
alternate exterior angles, they are congruent by the Alternate
Exterior Angles Theorem (Thm. 3.3). Because angle 2∠ and 3∠ are
consecutive angles, they are supplementary by the Consecutive
Interior Angles Theorem (Thm. 3.4).
7. 1 110 , 2 70 ;m m∠ = ° ∠ = ° Because ( ) ( )3 5 4 30 ,x x+ °
= − ° the value of x is 35. So, ( )3 5 110x + ° = ° and ( )4 30 110
.x − ° = ° By the Corresponding Angles Theorem (Thm. 3.1),
1 110 .m∠ = ° By the Linear Pair Postulate (Post 2.8), 2 70 .m∠
= °
STATEMENTS REASONS
1. p q 1. Given
2. 1 2 180m m∠ + ∠ = ° 2. Linear Pair Postulate (Post. 2.8)
3. 2 3 180m m∠ + ∠ = ° 3. Consecutive Interior Angles Theorem
(Thm. 3.4)
4. 1 3∠ ≅ ∠ 4. Congruent Supplements Theorem (Thm. 2.4)
STATEMENTS REASONS
1. 1 2∠ ≅ ∠ 1. Given
2. 1 3∠ ≅ ∠ 2. Vertical Angles Congruence Theorem (Thm. 2.6)
3. 2 3∠ ≅ ∠ 3. Transitive Property of Angle Congruence (Thm.
2.2)
12 p
q
t
3
Answers
Geometry Copyright © Big Ideas Learning, LLC Answers All rights
reserved. A26
8. 3, 5, 6, 7, 9,∠ ∠ ∠ ∠ ∠ and 10;∠ Because 1∠ and 3∠ are
supplementary to 2∠ by the Consecutive
Interior Angles Theorem (Thm. 3.4), 1 3∠ ≅ ∠ by the Congruent
Supplements Theorem (Thm. 2.4).
1 5∠ ≅ ∠ and 1 7∠ ≅ ∠ by the Alternate Interior Angles Theorem
(Thm. 3.3). 1 6∠ ≅ ∠ by the Vertical Angles Congruence Theorem
(Thm. 2.6). Because 3 9∠ ≅ ∠ by the Vertical Angles Congruence
Theorem (Thm. 2.6), 1 9∠ ≅ ∠ by the Transitive Property of Angle
Congruence (Thm. 2.2). Because 5 10∠ ≅ ∠ by the Vertical Angles
Congruence Theorem (Thm. 2.6),
1 10∠ ≅ ∠ by the Transitive Property of Angle Congruence (Thm.
2.2).
3.2 Enrichment and Extension 1. 65, 60x y= = 2. 13, 12x y= =
3.
4. 1 35 ,m∠ = ° 2 145 ,m∠ = ° 3 111 ,m∠ = °4 69 ,m∠ = ° 5 111
,m∠ = ° 6 69 ,m∠ = °7 145 ,m∠ = ° 8 35 ,m∠ = ° 9 69 ,m∠ = °10 111
,m∠ = ° 11 69 ,m∠ = ° 12 111 ,m∠ = °13 76 ,m∠ = ° 14 104 ,m∠ = ° 15
76 ,m∠ = °16 104 ,m∠ = ° 17 104 ,m∠ = ° 18 76 ,m∠ = °19 104 ,m∠ = °
20 76m∠ = °
5. 1 100 ,m∠ = ° 2 80 ,m∠ = ° 3 80 ,m∠ = °4 100 ,m∠ = ° 5 100
,m∠ = ° 6 56 ,m∠ = °7 24 ,m∠ = ° 8 24 ,m∠ = ° 9 56 ,m∠ = °10 100
,m∠ = ° 11 156 ,m∠ = ° 12 24 ,m∠ = °13 24 ,m∠ = ° 14 156 ,m∠ = ° 15
124 ,m∠ = °16 56 ,m∠ = ° 17 124 ,m∠ = ° 18 56 ,m∠ = °19 100 ,m∠ = °
20 80 ,m∠ = ° 21 100 ,m∠ = °22 80 ,m∠ = ° 23 156 ,m∠ = ° 24 24 ,m∠
= °25 24 ,m∠ = ° 26 156 ,m∠ = ° 27 100 ,m∠ = °28 56 ,m∠ = ° 29 24
,m∠ = ° 30 24 ,m∠ = °31 56 ,m∠ = ° 32 100m∠ = °
3.2 Puzzle Time GEOMETRY
3.3 Start Thinking
120°; 60° and 120°, respectively; The angles are the same as the
shopping mall sidewalks because they are parallel to them.
3.3 Warm Up 1. 10, 12x y= = 2. 5, 3x y= =
3.3 Cumulative Review Warm Up 1. 2 1m m∠ = ∠ 2. GH HJ+ 3. 4
GH•
3.3 Practice A 1. 44;x = Lines s and t are parallel when the
marked alternate exterior angles are congruent.
( ) ( )3 8 2 103 24 2 20
44
x x
x xx
− ° = + °
− = +=
A D
B C
STATEMENTS REASONS
1. ,AB DC AD BC 1. Given
2. A∠ and B∠ are supplementary.
2. Consecutive Interior Angles Theorem (Thm. 3.4)
3. B∠ and C∠ are supplementary.
3. Consecutive Interior Angles Theorem (Thm. 3.4)
4. 180m A m B∠ + ∠ = ° 4. Definition of supplementary angles
5. 180m B m C∠ + ∠ = ° 5. Definition of supplementary angles
6. m B m C∠ + ∠
6. Substitution
7. m A m C∠ = ∠ 7. Subtraction Property of Equality
8. A C∠ ≅ ∠ 8. Definition of congruent angles
m A m B∠ + ∠ =
60°
walkways
ShoppingMall
streets
60°
Answers
Copyright © Big Ideas Learning, LLC Geometry All rights
reserved. Answers
A27
2. 18;x = Lines s and t are parallel when the marked consecutive
interior angles are supplementary. ( )4 12 120 180
4 108 1804 72
18
x
xxx
− ° + ° = °
+ ===
3. yes; Corresponding Angles Converse (Thm. 3.5)
4. no
5. This diagram shows that the vertical angles are congruent,
and we do not have enough information to prove that .m n
6.
7. no; The labeled angles must be congruent to prove the wings
are parallel.
3.3 Practice B 1. 12;x = Lines s and t are parallel when the
marked alternate exterior angles are congruent.
( ) ( )4 16 7 2036 312
x x
xx
+ ° = − °
==
2. 26;x = Lines s and t are parallel when the marked consecutive
interior angles are supplementary. ( ) ( )2 15 3 20 180
2 30 3 20 1805 50 180
5 13026
x x
x xx
xx
+ ° + + ° = °
+ + + =+ =
==
3. yes; Alternate Exterior Angles Converse (Thm. 3.7)
4. yes; Consecutive Interior Angles Converse (Thm. 3.8)
5. a. yes; Lines a and b are parallel by the Alternate Interior
Angles Converse (Thm. 3.6). Lines b and c are parallel by the
Alternate Exterior Angles Converse Theorem (Thm. 3.7). Line c and d
are parallel by the Corresponding Angles Converse (Thm. 3.5). Lines
b and c are parallel by the Alternate Exterior Angles Converse
(Thm. 3.7). By the Transitive Property of Parallel Lines (Thm.
3.9), all the lines of latitude are parallel. b. no; There is not
enough information to prove that
the lines of longitude are parallel.
6. a. 27, 13, 9;x y z= = = Lines p and q are parallel when the
marked alternate exterior angles are congruent.
( ) ( )3 1 4 303 3 4 30
27
x x
x xx
− ° = − °
− = −=
Lines q and r are parallel when the marked corresponding angles
are congruent.
( ) ( )( )4 30 6
4 27 30 6
78 613
x y
y
yy
− ° = °
− =
==
The angles 6 y° and ( )6 8z + ° form a linear pair, so they are
supplementary.
( )( ) ( )
6 6 8 180
6 13 6 8 180
78 6 48 1806 54
9
y z
z
zzz
° + + ° = °
+ + =
+ + ===
b. yes; Because ( )3 1 78 and 6 78 ,x y− ° = ° ° = °lines p and
q are parallel by the Alternate Exterior Converse (Thm. 3.7).
STATEMENTS REASONS
1. 1 and 2∠ ∠ are supplementary.
1. Given
2. 2 and 3∠ ∠ are supplementary.
2. Linear Pair Postulate (Post 2.8)
3. 1 3∠ ≅ ∠ 3. Congruent Supplements Theorem (Thm. 2.4)
4. p q 4. Corresponding Angles Converse (Thm. 3.5)
Answers
Geometry Copyright © Big Ideas Learning, LLC Answers All rights
reserved. A28
7.
3.3 Enrichment and Extension 1. 78°
2.
STATEMENTS REASONS
1. 1 2∠ ≅ ∠ 1. Given
2. c d 2. Alternate Exterior Angles Converse (Thm. 3.7)
3. 2 3∠ ≅ ∠ 3. Given
4. a b 4. Alternate Interior Angles Converse (Thm. 3.6)
5. 3 4∠ ≅ ∠ 5. Corresponding Angles Theorem (Thm. 3.1)
6. 1 4∠ ≅ ∠ 6. Transitive Property of Angle Congruence (Thm.
2.2)
STATEMENTS REASONS
1. AC is parallel to .FG BD is the bisector of .CBE∠ DE is the
bisector of .BEG∠
1. Given
2. CBE BEF∠ ≅ ∠ 2. Alternate Interior Angles Theorem (Thm.
3.2)
3. m CBEm BEF
∠= ∠
3. Properties of Angle Congruence (Thm. 2.2)
4. ABE BEG∠ ≅ ∠ 4. Alternate Interior Angles Theorem (Thm.
3.2)
5. m ABEm BEG
∠= ∠
5. Properties of Angle Congruence (Thm. 2.2)
6. 180CBE ABE∠ + ∠
= ° 6. Definition of linear
pair
7. 180CBE BEG∠ + ∠
= ° 7. Substitution
8. 12 CBE DBE∠ = ∠ 8. Definition of angle bisector
9. 12 BEG BED∠ = ∠ 9. Definition of angle bisector
10.
( )
1 12 2
12 180
CBE BEG∠ + ∠
= °
10. Multiplication Property of Equality
11. 1 12 290
CBE BEG∠ + ∠
= °
11. Simplify
12. 90
DBE BED∠ + ∠= °
12. Substitution
13. 180
m DBE m BEDm EDB
∠ + ∠+ ∠ = °
13. Property of triangles
14. 180 90 EDB° = ° + ∠ 14. Substitution
15. 90 EDB° = ∠ 15. Subtraction Property of Equality
Answers
Copyright © Big Ideas Learning, LLC Geometry All rights
reserved. Answers
A29
3. a. one line b. an infinite number of lines c. one plane
4. a. 137° b. 71° c. 137° d. 43° e. 71°
5.
3.3 Puzzle Time BECAUSE HE WANTED TO SEE TIME FLY
3.4 Start Thinking Sample answer: framing square and chalk line;
A framing square ensures cuts made with saws are precise. The chalk
line helps builders keep a horizontal surface when needed.
3.4 Warm Up 1. 25 cm 2. 33 cm
3. 478.5 cm2 4. 46 cm 5. 7 cm
3.4 Cumulative Review Warm Up
1. Given AB CD≅ , prove CD AB≅
2. Given A∠ , prove A A∠ ≅ ∠
3.4 Practice A 1. about 5.7 units
2.
3. none; The only thing that can be concluded from the diagram
is that n⊥ and .m p⊥ In order to say that the lines are parallel,
you need to know something about the intersections of and p or m
and .n
STATEMENTS REASONS
1. CA ED
45m FED∠ = ° 1. Given
2. ABE∠ and DEB∠ are supplementary
2. Consecutive Interior Angles Theorem (Thm. 3.4)
3. 180
m ABE m DEB∠ + ∠= °
3. Definition of supplementary angles
4. 45180
m ABE∠ + °= °
4. Substitution Property of Equality
5. 135m ABE∠ = ° 5. Subtraction Property of Equality
6. 135m FBC∠ = ° 6. Vertical Angles Congruence Theorem (Thm.
2.6)
7. 45m GCA∠ = ° 7. Given
8. 135 45 180° + ° = ° 8. Addition
9. 180
m FBC m GCA∠ + ∠= °
9. Substitution Property of Equality
10. andFBC GCA∠ ∠ are supplementary.
10. Definition of supplementary angles.
11. EF CG
11. Consecutive Interior Angles Converse Theorem (Thm. 3.8)
STATEMENTS REASONS
1. AB CD≅ 1. Given
2. AB CD= 2. Definition of congruent segments
3. CD AB= 3. Symmetric Property of Equality
4. CD AB≅ 4. Definition of congruent segments
STATEMENTS REASONS
1. A∠ 1. Given
2. m A m A∠ = ∠ 2. Reflexive Property of Angle Measures
3. A A∠ ≅ ∠ 3. Definition of congruent angles
m
P
×
Answers
Geometry Copyright © Big Ideas Learning, LLC Answers All rights
reserved. A30
4. ||b c ; Because a b⊥ and a c⊥ , lines b and c are parallel by
the Lines Perpendicular to a Transversal Theorem (Thm. 3.12).
5.
6. no; There is only one perpendicular bisector that can be
drawn, but there is an infinite number of perpendicular lines.
7. || , || , ||w x w z x z ; Because w b⊥ and , ||x b w x⊥ by
the Lines Perpendicular to a
Transversal Theorem (Thm 3.12). Because w b⊥ and , ||z b w z⊥ by
the Lines Perpendicular to a Transversal Theorem (Thm 3.12).
Because ||w x and || , ||w z x z by the Transitive Property of
Parallel Lines Theorem (Thm. 3.9).
3.4 Practice B
1. 2 5 units
2. ||g h ; Because e g⊥ and e h⊥ , lines g and h are parallel by
the Lines Perpendicular to a Transversal Theorem (Thm. 3.12).
3. || , || , ||n m n m ; Because j ⊥ and j n⊥ , lines and n are
parallel by the Lines
Perpendicular to a Transversal Theorem (Thm. 3.12). Because k m⊥
and k n⊥ , lines m and n are also parallel by the Lines
Perpendicular to a Transversal Theorem (Thm. 3.12). Because || n
and ||m n , lines and m are parallel by the Transitive Property of
Parallel Lines Theorem (Thm. 3.9).
4. yes; Because || ,e f a e⊥ and c e⊥ , lines a and c are
perpendicular to line f by the Perpendicular Transversal Theorem
(Thm. 3.11). Because , , ,a f b f c f⊥ ⊥ ⊥ and ,d f⊥ by the Lines
Perpendicular to a Transversal Theorem (Thm. 3.12) and the
Transitive Property of Parallel Lines (Thm. 3.9), lines a, b, c,
and d are all parallel to each other.
5.
6. 1 90 , 2 15 , 3 90 ,m m m∠ = ° ∠ = ° ∠ = ° 4 45 , 5 15 ;m m∠
= ° ∠ = ° 1 90 ,m∠ = ° because
it is vertical angles with a right angle, so it has the same
angle measure. 2 90 75 15 ,m∠ = ° − ° = ° because it is
complementary to the 75° angle.
3 90 ,m∠ = ° because it is marked as a right angle. 4 75 30 45
,m∠ = ° − ° = ° because together with
the 30° angle, the angles are vertical angles with the 75°
angle, so the angle measures are equal.
5 15 ,m∠ = ° because it is vertical angles with 2,∠so the angles
have the same measure.
7. no; You do not know anything about the relationship between
lines x and y or x and z.
STATEMENTS REASONS
1. 1 2∠ ≅ ∠ 1. Given
2. e h⊥ 2. Linear Pair Perpendicular Theorem (Thm. 3.10)
3. ||e f 3. Lines Perpendicular to a Transversal Theorem (Thm.
3.12)
4. ||e g 4. Transitive Property of Parallel Lines (Thm. 3.9)
STATEMENTS REASONS
1. 1 2∠ ≅ ∠ 1. Given
2. a c⊥ 2. Linear Pair Perpendicular Theorem (Thm. 3.10)
3. ||c d 3. Given
4. a d⊥ 4. Perpendicular Transversal Theorem (Thm. 3.9)
5. b d⊥ 5. Given
6. ||a b 6. Lines Perpendicular to a Transversal Theorem (Thm.
3.12)
Answers
Copyright © Big Ideas Learning, LLC Geometry All rights
reserved. Answers
A31
3.4 Enrichment and Extension
1.
2.
STATEMENTS REASONS
1. ; 3AC BC⊥ ∠ is complementary to 1∠
1. Given
2. 1∠ is complementary to 2∠
2. Definition of perpendicular lines
3. 1 290
m m∠ + ∠= °
3. Definition of complementary angles
4. 1 390
m m∠ + ∠= °
4. Definition of complementary angles
5. 1 21 3
m mm m
∠ + ∠= ∠ + ∠
5. Substitution
6. 2 3m m∠ = ∠ 6. Substitution Property of Equality
7. 3 2m m∠ = ∠ 7. Symmetric Property of Equality
8. 3 2∠ ≅ ∠ 8. Definition of congruent angles
STATEMENTS REASONS
1. AB bisects DAC∠ ; CB bisects ECA∠ 2 45
3 45mm
∠ = °∠ = °
1. Given
2. 2 1m m∠ = ∠ 2. Definition of angle bisector
3. 1 45m∠ = ° 3. Substitution
4. 1 2m mm DAC
∠ + ∠= ∠
4. Angle addition
5. 45 45m DAC° + °
= ∠
5. Substitution
6. 90 m DAC° = ∠ 6. Simplify
7. DAC∠ is a right angle
7. Definition of a right angle
8. DA AC⊥
8. Definition of perpendicular lines
9. 3 4m m∠ = ∠ 9. Definition of angle bisector
10. 4 45m∠ = 10. Definition of congruent angles
11. 3 4m mm ECA
∠ + ∠= ∠
11. Angle addition
12. 45 45m ECA° + °
= ∠
12. Substitution
13. 90 m ECA° = ∠ 13. Simplify
14. ECA∠ is a right angle.
14. Definition of a right angle
15. EC AC⊥
15. Definition of perpendicular lines
16. AD
is parallel to .CE
16. Lines Perpendicular to a Transversal Theorem (Thm. 3.12)
Answers
Geometry Copyright © Big Ideas Learning, LLC Answers All rights
reserved. A32
3.
4.
5. 7d = 6. 52
d =
7. 834
d = 8. 313
d =
3.4 Puzzle Time THE ADDER
3.5 Start Thinking
The lines 3y x= − and 2y x= + do not intersect; The line 5y x= −
+ intersects the line
3y x= − at the point ( )4, 1 and the line
2y x= + at the point 3 7, ;2 2
The angles are
right angles.
3.5 Warm Up 1.
2.
3.
STATEMENTS REASONS
1. , 1 3j ⊥ ∠ ≅ ∠ 1. Given
2. 2 390
m m∠ + ∠= °
2. Definition of complementary angles
3. 1 3m m∠ = ∠ 3. Definition of congruent angles
4. 2 190
m m∠ + ∠= °
4. Substitution
5. BED∠ is a right angle
5. Definition of a right angle
6. k m⊥ 6. Definition of perpendicular lines
STATEMENTS REASONS
1. m n⊥ 1. Given
2. 3∠ and 6∠ are complementary.
2. Definition of complementary angles
3. 3∠ and 4∠ are complementary.
3. Given
4. 4 6∠ ≅ ∠ 4. Congruent Complements Theorem (Thm. 2.5)
5. 4 5∠ ≅ ∠ 5. Vertical Angles Congruence Theorem (Thm. 2.6)
6. 5 6∠ ≅ ∠ 6. Transitive Property of Congruence (Thm 2.2)
y
x41−1
−2
4
y = x − 3
y = x + 2y = −x + 5
x
y
2
4
−2−6 6
y = 6x
x
y
2
4
−2−6 6
y = 4x + 2
x
y
41
2
−2
−1
y = x − 3
Answers
Copyright © Big Ideas Learning, LLC Geometry All rights
reserved. Answers
A33
4.
5.
6.
3.5 Cumulative Review Warm Up 1. Multiplication Property of
Equality
2. Subtraction Property of Equality
3. Reflexive Property of Equality for Real Numbers
4. Reflexive Property of Equality for Angle Measures
5. Transitive Property of Equality for Angle Measures
6. Symmetric Property of Segment Lengths
3.5 Practice A 1. ( )3.5, 1P 2. ( )0, 14.2P
3. perpendicular; Because
1 29 2 1,2 9
m m • = − = −
lines 1 and 2 are
perpendicular by the Slopes of Perpendicular Lines Theorem (Thm.
3.14).
4. neither; Because 1 24 5 1,5 4
m m • = =
lines 1
and 2 are neither parallel nor perpendicular.
5. 4 7y x= + 6. 6 9y x= − +
7. 1 83
y x= + 8. 3 8y x= −
9. 2 2 2.83≈ 10. 2 26 10.2≈
11. 7.5−
12. no; For a line with a slope between 0 and 1, the slope of a
line perpendicular to it would be negative.
13. ( )5, 4
3.5 Practice B 1. ( )1.5, 3Q = 2. ( )0, 3Q =
3. neither; Because ( )1 2 1 12 ,6 3m m • = − = −
lines
1 and 2 are neither parallel nor perpendicular.
4. 6 10y x= − − 5. 1 114 4
y x= − +
6. about 4.5 7. about 4.4
8. Sample answer: 5, 1b c= =
9. a. The slope is 2 2, where 1 0.m m− ≤ <
b. The slope is 3 3, where 1.m m ≥
c. The lines are perpendicular; They are perpendicular by the
Perpendicular Transversal Theorem (Thm. 3.11).
10. yes; Sample answer: The lines 12 and2
y x y x= = − have the same y-intercept
and the slopes are negative reciprocals.
11. 5 , 22
− −
3.5 Enrichment and Extension
1. 4 23 3
y x= − − 2. 18, 30a b= =
3. a. 3.62 b. 2.74 c. 3.62 17.8926y x= −
d. 0.276 1.412y x= −
x41−1
−4
2y
y = x − 223
1 4
2
x
y
−1
−2
x + 3y = − 43
y
1−4 −1
−2
3y = x + 2
x
Answers
Geometry Copyright © Big Ideas Learning, LLC Answers All rights
reserved. A34
4. ; parallel: ; perpendicular:a ax by y xb b a
− = − =
5. 14, 102
k y x= − = − +
6. k can have any value, 2 5y x= −
7. a. ae db≠
b. , 0, 0a d b eb e
− = − ≠ ≠
8. a. Sample answer: ( ) ( ) ( )4, 4 , 4, 4 , 0, 2− b. Sample
answer:
4, 4, 0, 4, 2x x x y y= − = = = =
c. ( ) ( ), , , , 0,2yy y y y −
3.5 Puzzle Time DROP THE S
Cumulative Review 1. 0 2. 0 3. 13
4. 40 5. 5− 6. 132
7. 25 8. 29 9. 12
10. 4− 11. 84− 12. 3
13. 29 14. 58− 15. 6−
16. 20− 17. 24− 18. 14
19. 16− 20. 19− 21.
13
22. 4 23. 5− 24. 7
25. 9− 26. 74− 27. 52
28. 53 29. 92− 30.
25
31. 78 32. 32− 33.
53−
34. a. 11 A.M. b. 6.5 in. c. 3 P.M.
35. a. about $42.92 b. about $9.90 c. about $1.41
36. 16x = 37. 8x = 38. 6x = −
39. 35x = − 40. 1x = − 41. 8x =
42. 9x = 43. 49x = − 44. 11x = −
45. 12x = − 46. 11x = − 47. 3x =
48. 3x = − 49. 1x = − 50. 6x = −
51. 5x = 52. 3, 4m b= = −
53. 4, 5m b= − = 54. 34 , 7m b= = −
55. 56, 3m b= − = 56. 1, 5m b= =
57. 1, 3m b= − = 58. 1, 1m b= − = −
59. 2, 9m b= − = 60. 3, 8m b= − =
61. 2, 5m b= = − 62. 5, 87
m b= =
63. 23, 4m b= − = − 64. 10x
65. a. 8 5 3− = b. 8 5.5 2.5− = c. Company A is 3 minutes
faster. Company B is
2.5 minutes faster.
66. 9.4 67. 7.1 68. 20.4
69. 10.2 70. 16.4 71. 15.8
72. 16.3 73. 6.7 74. 18.4
75. 12.4 76. 7.8 77. 9.2
78. ( )2, 5.5− 79. ( )8, 1− 80. ( )3.5, 1− −
81. ( )2.5, 4 82. ( )2.5, 9.5− 83. ( )3, 0.5−
84. ( )5.5, 2− 85. ( )0.5, 7− 86. ( )5.5, 1.5−
87. ( )0.5, 1.5− − 88. ( )1.5, 0.5−
89. ( )0.5, 0.5−
90. a. each individual visit b. each individual visit c. 5 or
more visits
Answers
Copyright © Big Ideas Learning, LLC Geometry All rights
reserved. Answers
A35
91. a. $7.80 b. $9.70 c. 7 lb
92. 2 3y x= − 93. 3 27y x= − −
94. 1 12 25y x= + 95. 15 4y x= +
96. 14 2y x= − + 97. 8 93y x= +
98. 4 29y x= − + 99. 2 15 5y x= − +
100. 13 11y x= − 101. 12 3y x= − +
102. 19 10y x= − 103. 2 2y x= − +
104. 3 22y x= + 105. 7y = −
106. 13 3y x= − + 107. 46x =
108. 136x = 109. 28x =
110. 19x = 111. 35x = 112. 21x =
Chapter 4 4.1 Start Thinking
Translate the original triangle 2 units down; Each ordered pair
for A B C′ ′ ′Δ contains y-coordinates that are two less than those
of ;ABCΔ When identifying a translation, you can compare the x- and
y-values to determine what happens if the figure is plotted.
4.1 Warm Up
1. 2.
( 2, 2)P′ − (0, 1)P′
3. 4.
(4, 0)P′ ( 4, 5)P′ −
5. 6.
(6, 0)P′ (4, 4)P′
4.1 Cumulative Review Warm Up 1.
1 2p q∠ ≅ ∠
GivenProve
x
y4
2
−2
−4
42−4 −2
P
P′
x
y4
2
−2
−4
42−4 −2
P
P′
x
y4
2
−2
−4
42−4 −2
P
P′
x
y
4
2
−2
42−4 −2
PP′
x
y4
2
−2
−4
4 6−2 PP′
x
y4
2
−2
−4
42 6−2P
P′
1
2
3q
p
t
STATEMENTS REASONS
1. p q 1. Given
2. 1 3∠ ≅ ∠ 2. Corresponding Angles Theorem (Thm. 3.1)
3. 3 2∠ ≅ ∠ 3. Vertical Angles Congruence Theorem (Thm. 2.6)
4. 1 2∠ ≅ ∠ 4. Transitive Property of Angle Congruence (Thm.
2.2)
x
y4
2
−2
42−4
A
C
B
B′C′
A′