Geometry RBC A answers - Mr. Reevesmrdreeves.weebly.com/uploads/2/2/9/8/22981178/unit_3_practice... · 3. CG 4. AB BC ... Because angle ∠2 and ∠3 are consecutive angles, they

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


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°


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


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


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).


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= =


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.


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


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


7.


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)


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


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


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


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


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


A31


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∠ + ∠= °


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⊥


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.


15. EC AC⊥


16. AD

is parallel

to .CE

16. Lines Perpendicular to a Transversal Theorem (Thm. 3.12)

Answers


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∠ + ∠= °


3. 1 3m m∠ = ∠ 3. Definition of congruent angles

4. 2 1

90

m m∠ + ∠= °

4. Substitution

5. BED∠ is a right angle


6. k m⊥ 6. Definition of perpendicular lines

STATEMENTS REASONS

1. m n⊥ 1. Given

2. 3∠ and 6∠ are complementary.


3. 3∠ and 4∠ are complementary.

3. Given

4. 4 6∠ ≅ ∠ 4. Congruent Complements Theorem (Thm. 2.5)


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


A33

4.

5.

6.


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 − −


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


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


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′


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)



x

y4

2

−2

42−4

A

C

B

B′C′

A′

Documents

Geometry RBC A answers - Mr. Reevesmrdreeves.weebly.com/uploads/2/2/9/8/22981178/unit_3_practice... · 3. CG 4. AB BC ... Because angle ∠2 and ∠3 are consecutive angles, they