name · 2020. 3. 30. · name pearsonrealize.com 2-3 additional practice parallel lines and...

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2-1 Reteach to Build UnderstandingParallel Lines

1. Use the diagrams to fill in the blanks. Determine whether the pair of angles are congruent or supplementary and find the measure of the angle.

Same-Side Interior Angles

Alternate Interior Angles

Corresponding Angles

Alternate Exterior Angles

1

2

1

2

1

2

1

2

∠1 and ∠2 are .

∠1 and ∠2 are .

∠1 and ∠2 are .

∠1 and ∠2 are .

If m∠1 = 100°, then m∠2 = .

If m∠1 = 75°, then m∠2 = .

If m∠1 = 82°, then m∠2 = .

If m∠1 = 77°, then m∠2 = .

2. A student said that m∠1 = 80°. What error did the student likely make? What is m∠1?

The student likely confused same-side interior angles and alternate interior angles because they are both interior. m∠1 = 100°

3. Complete the two-column proof to prove the Alternate Exterior Angles Theorem.

Given: k || ℓ

Prove: ∠1 ≅ ∠7

Statements Reasons

1) k || ℓ 1) Given

2) ∠1 ≅ ∠5 2)

3) ∠5 ≅ ∠7 3) Vertical Angles Theorem

4) ∠1 ≅ ∠7 4) Transitive Property of Congruence

100°

1

1 57

kℓ

m

Corresponding Angles Theorem

supplementary congruent congruent congruent

80° 75° 82° 77°

enVision™ Geometry • Teaching Resources

4 31 2 5 6

78t

lt

128°1 23

4 56

7

a

b

c

32 65

7814

a

x y

A B C D

F

E

K

LM1

2

3

53°

G H I

J


2-1 Additional PracticeParallel Lines

Use the figure for Exercises 1–4. Identify all pairs of each type of angle.

1. corresponding angles ∠ 1 and ∠ 5; ∠ 2 and ∠ 6; ∠ 3 and ∠ 7; ∠ 4 and ∠ 8

2. same-side interior angles ∠ 2 and ∠ 5; ∠ 3 and ∠ 8

3. alternate interior angles ∠ 2 and ∠ 8; ∠ 3 and ∠ 5

4. alternate exterior angles ∠ 1 and ∠ 7; ∠ 4 and ∠ 6

Use the figure for Exercises 5 and 6.

5. Which angles are supplementary to the given angle? ∠ 1, ∠ 2, ∠ 4, ∠ 6

6. Which angles are congruent to the given angle? ∠ 3, ∠ 5, ∠ 7

7. Complete the two-column proof.

Given: x ∥ y

Prove: ∠ 3 ≅ ∠ 5

Statements Reasons

1) x ∥ y

2) m∠3 + m∠8 = 180°

3) m∠5 + m∠8 = 180°4) m∠3 + m∠8 = m∠5 + m∠85) m∠3 = m∠56) ∠3 ≅ ∠5

1) Given2) Same-Side Interior Angle

Postulate3) Angle Addition Postulate4) Transitive Property of Equality

5) Subtraction Property of Equality

6) Definition of congruence

8. In the figure, EH ∥ AI and AI ∥ CJ.

a. What is m ∠ 1? Explain. 127°; by the Same-Side Interior Angles Postulate, m ∠ 1 + 53° = 180°, so m ∠ 1 = 180° − 53° = 127°.

b. What is m ∠ 3? Explain. 53°; by the Corresponding Angles Theorem, ∠ CMG ≅ ∠ 2 and ∠ 2 ≅ ∠ 3, so ∠ 3 ≅ ∠ CMG by the Transitive Property of Congruence, and m ∠ CMG = 53°, so m ∠ 3 = 53°.



2-2 Reteach to Build UnderstandingProving Lines Parallel

1. Match each figure with the statement that proves ℓ || m.

2. A student said that when x = 6, a || b. What error did the student likely make? For what value of x is a || b?

The student set the given angles equal when they are actually supplements. a || b when x = 14.

3. Complete the flow proof of the Converse of the Corresponding Angles Theorem. Fill in the blanks.

Given: ∠2 ≅ ∠3

Prove: c || d

c

d1

2

3

∠2 ≅ ∠3

Given

∠1 ≅ ∠3

TransitiveProp. of ≅

Converse of theCorr. ∠s Thm.

∠1 ≅ ∠2Vertical Angles

Theorem

c ∣∣ d

Converse of the Alternate Exterior Angles Theorem

Converse of the Corresponding Angles Theorem

Converse of the Same-Side Interior Angles Postulate

ℓ

∠1 ≅ ∠4

1

4 m

ℓ

m∠3 + m∠5 = 180°3

5 m

ℓ

∠1 ≅ ∠6

1

6m

a

b

(5x + 12)°(7x)°


3

67

4

21

8

5

d

es

t

1

2

3

4

a

b

c

d


2-2 Additional PracticeProving Lines Parallel

Use the figure for Exercises 1–4. Using the given information, which lines can you conclude are parallel? State the theorem or postulate that justifies each answer.

1. ∠1 ≅ ∠4 line s and line t; Alternate Exterior Angles Theorem

2. ∠2 ≅ ∠3 line d and line e; Corresponding Angles Theorem

3. ∠6 ≅ ∠7 line d and line e; Alternate Interior Angles Theorem

4. m∠5 + m∠8 = 180° line s and line t; Same-Side Interior Angles Postulate

5. Complete the flow proof shown. Fill in the blanks.

Given: ∠1 and ∠4 are supplementary; c ∥ d .

Prove: a ∥ b

c || d

a || b

m∠3 + m∠4 = m∠1 + m∠4

m∠1 = m∠3

m∠1 + m∠4 = 180°∠1 and ∠4are suppl.

∠1 ≅ ∠3

∠2 ≅ ∠3

C || d

Given

Given

Same-side interior∠s are suppl.

m∠3 + m∠4 = 180°

Vertical ∠s are ≅.

Substitution

Subtraction Propertyof Equality

TransitiveProperty of ≅

Converse ofCorresponding

Angles Theorem

Definition ofcongruent ∠s

Definition of suppl. ∠s

∠1 ≅ ∠2

In Exercises 6 and 7, for what value of x is l ∥ m ? Which theorem or postulate justifies your answer in each case?

6. (2x + 2)°

(3x – 63)°ℓ

m

7. (2x + 20)°

(6x + 24)°

m

ℓ

x = 17; Same-Side Interior Angles Postulate

x = 65; Alternate Exterior Angles Theorem



2-3 Reteach to Build UnderstandingParallel Lines and Triangle Angle Sums

1. Use the diagrams to fill in the blanks.

Interior Angle Measures Exterior Angle Measure

54°

48°

A

C

B 40°

59°

A

BCD

m∠A + m∠B + m∠C = 180° m∠DCA = m∠A + m∠B

+ m∠B + = 180° m∠DCA = +

m∠B + = 180° m∠DCA =

m∠B =

2. From the figure shown, Abby concludes that m∠1 = 85°. What is Abby’s error? How would you correct the error?

The sum of the interior angles is 180°, so the measure of angle 2 is 85°, but the measure of angle 1 is not. The measure of angle 1 is the sum of 62° and 33°, so m∠1 = 95°.

3. Complete the sentences to find the value of x.

The measure of the angle of the triangle equals the sum of the measures of its two

angles.

So the angle equals the sum of the angle and the x° angle.

So, the x° angle is the of the 117° angle and the 29° angle.

Therefore, x = .

21

62°

33°

29°

117°x°

40°48°99°102°

78°

117° 29°

88

59°54°

exterior

difference

remote interior



2-3 Additional PracticeParallel Lines and Triangle Angle Sums

For Exercises 1–6, find the value of each variable.

1. 92° x°

63°

2. x°

87°

20°

3.

x°

80°

40°

4.

85°

28°15°

x° y° z°

5.

43°

24°

57°x° y° z°

6. 11°

84°

50°z°

y°x°

For Exercises 7–12, find the value of x for each figure.

7.

70°

38°

x°

8.

39°

94°

x°

9. 43°

81°

x°

10. 19°

81°

x°

11.

12.

41°

36°

x°

Use the figure for Exercises 13–15. Find the measure of the given angle.

43°112°

33°

71°

c

ba

13. a 35° 14. b 74° 15. c 66°

115°

41°

x°

x = 25

x = 67, y = 113, z = 52

x = 73

x = 56, y = 81, z = 99

x = 60

x = 108x = 133 x = 124

x = 62 x = 74 x = 77

x = 35, y = 134, z = 46



2-4 Reteach to Build UnderstandingSlopes of Parallel and Perpendicular Lines

1. Match each set of lines with the correct statement.

Line a is parallel to line b.

a || b

Line a is perpendicular to line b.

a ⊥ b

2. Fill in the blanks to find the equation of the line perpendicular to y = 1 __

3 x − 2 through the point (1, 4).

The slope of the given line is . Perpendicular lines have slopes with a product of , so the slope of the perpendicular line is . To write the equation of the perpendicular line, use the form to solve for the y-intercept.

y = mx + b

= −3 ( ) + b

b =

The equation of the line perpendicular to y = 1 __ 3 x − 2 passing through the point

(1, 4) is .

3. Danielle says that the line perpendicular to y = 5x + 9 and passing through the point (3, 4) is y = 5x − 11. What is Danielle’s error? How would you correct the error?

The slopes of the two lines are equal, so the line y = 5x − 11 is parallel to the line y = 5x + 9. The line perpendicular to y = 5x + 9 passing through the point (3, 4) is y = − 1 __

5 x + 23 __

5 .

ya

xb

ya

x

b

The slope of line a times the slope of line b equals −1.

The slope of line a equals the slope of line b.

1 __ 3

slope-intercept−1 −3

17

y = −3x + 7

4


f(x)

g(x)

2 4 6−4−6 −2

2

−2

−4

4

6

xO

y

s

t

uv

q r


2-4 Additional PracticeSlopes of Parallel and Perpendicular Lines

1. A hand rail is installed along the stairs of a new house as shown in the figure. The table shows the distance, in inches, of the top rail f(x) and bottom rail g(x) from the floor for the middle of each numbered step x. Determine the slope of each rail. Are the top and bottom rails parallel?

f(x) g(x)x

1

2

3

9

16

23

43

50

57

The slope of the top rail is 7 and the slope of the bottom rail is 7. The slopes are the same, so the top and bottom rails are parallel.

Use the figure for Exercises 2–9. Determine whether each pair of lines are parallel or perpendicular. Write yes or no.

2. q and v, parallel yes

3. r and s, parallel no

4. r and t, parallel no

5. s and u, parallel yes

6. q and s, perpendicular no

7. q and v, perpendicular no

8. r and s, perpendicular yes

9. t and v, perpendicular no

Write the equations for the line parallel and the line perpendicular to the given line passing through the given point.

10. y = 2x + 7 ; (0, 1) parallel: y = 2x + 1 ; perpendicular: y = − 1 __ 2 x + 1

11. y = − 1 __ 3 x + 2 ; (3, 5) parallel: y = − 1 __

3 x + 6 ; perpendicular: y = 3x − 4

12. y = −5x − 1 __ 2 ; (–4, 2) parallel: y = − 5x − 18 ; perpendicular: y = 1 __

5 x + 14 __

5


name · 2020. 3. 30. · name pearsonrealize.com 2-3 additional practice parallel lines and...

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