Solving with Parallel Lines

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parallel: matching angles equal

consecutive interior: sum to 180°

perpendicular: 90°

What to Remember

Parallel − matching angles

If the lines are parallel, corresponding and alternate interior (and alternate exterior) angles are congruent. Set the expressions equal.

Example: 3x + 10 = 70, so x = 20.

Parallel − consecutive interior

Consecutive interior angles add to 180°.

Example: (4x + 10) + 70 = 180, so 4x + 80 = 180, x = 25.

Perpendicular

A right angle is 90°. If (2x + 16)° is marked with a square, then 2x + 16 = 90 and x = 37.

Two variables

You need two relationships: often a linear pair (sum 180°) and a corresponding or alternate pair (equal).

Level 1

Easy

One variable, one relationship.

Example

Lines a and b are parallel. Corresponding angles are 3x + 10 and 70. Find x.

Corresponding angles are congruent.

3x + 10 = 70

3x = 60

x = 20

Answer: 20

Lines a and b are parallel. Consecutive interior angles are 4x + 10 and 70. Find x.

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(4x + 10) + 70 = 180

4x + 80 = 180

4x = 100

x = 25

Answer: 25

A right angle is labeled (5x + 10)°. Find x.

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5x + 10 = 90

5x = 80

x = 16

Answer: 16

Lines a and b are parallel. Alternate interior angles are 2x + 8 and 40. Find x.

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2x + 8 = 40

2x = 32

x = 16

Answer: 16

Lines a and b are parallel. Corresponding angles are 6x − 12 and 4x + 8. Find x, then find the angle measure.

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6x − 12 = 4x + 8

2x = 20

x = 10

Angle = 4(10) + 8 = 48°

Answer: x = 10; 48°

Two lines are perpendicular. One of the four angles is (3x + 6)°. Find x.

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3x + 6 = 90

3x = 84

x = 28

Answer: 28

Level 2

Medium

Two expressions, or a right-angle pair.

Example

Lines a and b are parallel. Consecutive interior angles are 5x + 20 and 3x + 40. Find x and both angle measures.

(5x + 20) + (3x + 40) = 180

8x + 60 = 180

8x = 120

x = 15

5(15) + 20 = 95°

3(15) + 40 = 85°

95 + 85 = 180. Check.

Answer: x = 15; 95° and 85°

The diagram shows two perpendicular lines. The marked angles are (2x + 16)° and (4y + 10)°.

Find x and y.

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Each marked angle is 90°.

2x + 16 = 90

2x = 74

x = 37

4y + 10 = 90

4y = 80

y = 20

Answer: x = 37, y = 20

Lines a and b are parallel. Alternate exterior angles are 7x − 4 and 3x + 28. Find the angle measure.

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7x − 4 = 3x + 28

4x = 32

x = 8

Angle = 3(8) + 28 = 52°

Answer: 52°

Lines a and b are parallel. A linear pair on the top line is 2x + y and 3y, and the bottom corresponding angle to 2x + y is x + 4y.

Find x and y.

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Linear pair: (2x + y) + 3y = 180

2x + 4y = 180

x + 2y = 90

Corresponding: 2x + y = x + 4y

x = 3y

Substitute: 3y + 2y = 90

5y = 90

y = 18

x = 54

Answer: x = 54, y = 18

The top and bottom of the hourglass are parallel. m∠1 = 4a − 3b, m∠2 = 5a, m∠3 = 10b, and m∠4 = 55°.

Find a and b.

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∠2 and ∠3 are vertical, so 5a = 10b.

a = 2b

∠1 and ∠4 are corresponding, so 4a − 3b = 55.

4(2b) − 3b = 55

8b − 3b = 55

5b = 55

b = 11

a = 22

Answer: a = 22, b = 11

Lines a and b are parallel. Corresponding angles are x² + 5x − 36 and 3x + 12. Find x and the angle measure. Reject any value that makes an angle 0 or negative.

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x² + 5x − 36 = 3x + 12

x² + 2x − 48 = 0

(x + 8)(x − 6) = 0

x = −8 or x = 6

x = −8 gives 3(−8) + 12 = −12. Reject it.

x = 6

Angle = 3(6) + 12 = 30°

Answer: x = 6; 30°

Level 3

Hard

A quadratic, a two-variable system, or mixed relationships.

Example

Lines a and b are parallel. Consecutive interior angles are x² + 2x and 4x + 20. Find x and both angle measures. Reject any value that makes an angle 0 or negative.

(x² + 2x) + (4x + 20) = 180

x² + 6x + 20 = 180

x² + 6x − 160 = 0

(x + 16)(x − 10) = 0

x = −16 or x = 10

x = −16 makes 4x + 20 negative. Reject it.

x = 10

Angles: 120° and 60°

Answer: x = 10; 120° and 60°

The top and bottom of the hourglass are parallel. m∠1 = 3a + b, m∠2 = 4a, m∠3 = 8b, and m∠4 = 56°. Find a and b.

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Vertical: 4a = 8b, so a = 2b.

Corresponding: 3a + b = 56.

3(2b) + b = 56

6b + b = 56

7b = 56

b = 8

a = 16

Check: m∠2 = 4(16) = 64°, m∠3 = 8(8) = 64°.

m∠1 = 3(16) + 8 = 56°.

Answer: a = 16, b = 8

Lines a and b are parallel. A linear pair on the top line is x + 2y and 3x. The angle corresponding to x + 2y on the bottom line is 4y + 10. Find x and y.

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Linear pair: (x + 2y) + 3x = 180

4x + 2y = 180

2x + y = 90

Corresponding: x + 2y = 4y + 10

x = 2y + 10

2(2y + 10) + y = 90

4y + 20 + y = 90

5y = 70

y = 14

x = 2(14) + 10 = 38

Answer: x = 38, y = 14

Lines a and b are parallel. Corresponding angles are 2x + y and 50. Consecutive interior angles are 2x + y and x + 3y. Find x and y.

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Corresponding: 2x + y = 50

Consecutive interior: (2x + y) + (x + 3y) = 180

50 + x + 3y = 180

x + 3y = 130

From 2x + y = 50, y = 50 − 2x.

x + 3(50 − 2x) = 130

x + 150 − 6x = 130

−5x = −20

x = 4

y = 50 − 8 = 42

Answer: x = 4, y = 42

Two perpendicular lines have marked angles (3x + 6)° and (5y − 10)°. Find x and y.

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3x + 6 = 90

3x = 84

x = 28

5y − 10 = 90

5y = 100

y = 20

Answer: x = 28, y = 20

Lines a and b are parallel. Corresponding angles are x² + 3x − 40 and 2x + 20. Find x and the angle measure. Reject any value that makes an angle 0 or negative.

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x² + 3x − 40 = 2x + 20

x² + x − 60 = 0

(x + 10)(x − 6) = 0

x = −10 or x = 6

x = −10 gives 2(−10) + 20 = 0. Reject a 0° angle.

x = 6

Angle = 2(6) + 20 = 32°

Answer: x = 6; 32°

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