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 ; 2 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 11
48 6
8
x
x
x
° = − °
==
4. 10;
( )2 142 180
2 9 142 180
2 18 142 180
2 160 180
2 20
10
m
x
x
x
x
x
∠ + ° = °
+ ° + ° = °
+ + =+ =
==
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 ; 2 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 ; 2 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 8
32 3
32 2
16
x x
x x
x
x
+ ° = − °
+ ===
4. 51; ( ) ( )227 3 25 180
32
18 3 25 1803
117 180
351
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. 9, 12x y= = 2. 10, 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 10
3 24 2 20
44
x x
x x
x
− ° = + °
− = +=
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
4 108 180
4 72
18
x
x
x
x
− ° + ° = °
+ ===
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 20
36 3
12
x x
x
x
+ ° = − °
==
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 20 180
5 50 180
5 130
26
x x
x x
x
x
x
+ ° + + ° = °
+ + + =+ =
==
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 30
3 3 4 30
27
x x
x x
x
+ ° = − °
− = −=
Lines q and r are parallel when the marked corresponding angles are congruent.
( ) ( )( )4 30 6
4 27 30 6
78 6
13
x y
y
y
y
− ° = °
− =
==
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 48 180
6 54
9
y z
z
z
z
z
° + + ° = °
+ + =
+ + ===
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 CBE
m BEF
∠= ∠
3. Properties of Angle Congruence (Thm. 2.2)
4. ABE BEG∠ ≅ ∠ 4. Alternate Interior Angles Theorem (Thm. 3.2)
5. m ABE
m BEG
∠= ∠
5. Properties of Angle Congruence (Thm. 2.2)
6.
180
CBE ABE∠ + ∠= °
6. Definition of linear pair
7.
180
CBE 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 2
90
CBE BEG∠ + ∠
= °
11. Simplify
12.
90
DBE BED∠ + ∠= °
12. Substitution
13.
180
m DBE m BED
m 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. 45
180
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. 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 2
90
m m∠ + ∠= °
3. Definition of complementary angles
4. 1 3
90
m m∠ + ∠= °
4. Definition of complementary angles
5. 1 2
1 3
m m
m 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 m
m DAC
∠ + ∠= ∠
4. Angle addition
5. 45 45
m 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 m
m ECA
∠ + ∠= ∠
11. Angle addition
12. 45 45
m 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. 5
2d =
7. 8
34d = 8. 3
13d =
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 3
90
m m∠ + ∠= °
2. Definition of complementary angles
3. 1 3m m∠ = ∠ 3. Definition of congruent angles
4. 2 1
90
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 21 1
2 ,6 3
m m • = − = −
lines
1 and 2 are neither parallel nor perpendicular.
4. 6 10y x= − − 5. 1 11
4 4y x= − +
6. about 4.5 7. about 3.4
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 1
2 and2
y x y x= = − have the same y-intercept
and the slopes are negative reciprocals.
11. 5, 2
2 − −
3.5 Enrichment and Extension
1. 4 2
3 3y 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 b
y y xb b a
− = − =
5. 1
4, 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. ( ) ( ), , , , 0,2
yy 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 15 2
5y x= + 95. 15
4y x= +
96. 14 2y x= − + 97. 8 93y x= +
98. 4 29y x= − + 99. 2 15 5
y 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 2
p q
∠ ≅ ∠Given
Prove
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 6−2
P
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′