We included **HMH Into Math Grade 7 Answer Key PDF** **Module 7 Lesson 5 Two-Step Equations to Find Angle Measures **to make students experts in learning maths.

## HMH Into Math Grade 7 Module 7 Lesson 5 Answer Key Two-Step Equations to Find Angle Measures

I Can identify angle relationships, and use them to write and solve equations.

**Step It Out**

You previously learned that a right angle measures 90°. In this lesson, you will also work with pairs of angles called complementary angles and supplementary angles.

Connect to Vocabulary

Two angles whose measures have a sum of 90° are called complementary angles. Two angles whose measures have a sum of 1800 are called supplementary angles.

Question 1.

Use the diagram for Parts A-H.

A. Name the right angles. ∠ ____________ and ∠ ____________

Answer:

∠CBE and ∠CBA

Explanation:

The right angles are ∠CBE and ∠CBA.

B. Name a pair of complementary angles. ∠ ____________ and ∠ ____________

Answer:

∠DBE and ∠CBD.

Explanation:

A pair of complementary angles. ∠DBE and ∠CBD.

C. What is the measure of ∠ABE? ____________

Answer:

∠180°

Explanation:

The measure of ∠ABE is ∠180°

D. Name a pair of supplementary angles.

∠____________ and ∠ _____________

Answer:

∠ABD and ∠CBE.

Explanation:

A pair of supplementary angles is ∠ABD and ∠CBE.

E. If the measure of ∠CBD is equal to (5x)° and the measure of ∠DBE is 40°, use what you know about complementary angles to write an equation to solve for x. Then solve it.

+ =

x =

Answer:

x = 10°

Explanation:

Given,

∠CBD = 5x° and ∠DBE is 40°

∠CBD + ∠DBE = 90°

5x° + 40° = 90°

5x° = 90 – 40

5x° = 50°

x = 50°÷ 5

x = 10°

F. If the measure of ∠ABD is (5y + 15)° and the measure of ∠DBE is 40°, write an equation and solve for y.

+ =

y =

Answer:

y = 25°

Explanation:

Given,

∠ABD = (5y + 15)° and ∠DBE is 40°

∠ABD + ∠DBE = 180°

(5y + 15) + 40° = 180°

5y + 15 = 180 – 40

5y + 15 = 140

5y = 140 – 15

5y = 125°

y = 125° ÷ 5

y = 25°

**Turn and Talk** If ∠ABD were the only known angle measure, how could you determine the measures of the other angles in the diagram?

Answer:

∠ABD was given, ∠ABC = 90° we can find ∠DBE

∠ABD + ∠ABC + ∠DBE = 180°

With these measures of angles, we can find angles in the diagram.

Question 2.

Use Figure 1 for Parts A-C and Figure 2 for Parts D-F.

Connect to Vocabulary

Vertical angles are opposite angles formed by intersecting lines. Vertical angles are congruent.

A. Name two pairs of vertical angles.

∠____________ and ∠ _____________

∠ ___________ and ∠ ____________

Answer:

∠FGJ and ∠HGI

∠FGH and ∠JGI

Explanation:

The two pairs of vertical angles are ∠FGJ and ∠HGI, ∠FGH and ∠JGI

B. Vertical angles (are / are not) congruent.

Answer:

Vertical angles are congruent.

C. Given ∠FGH measures (6x – 24)° and ∠JGl measures 96°, write an equation that can be used to determine the value of x. Solve for x.

= and x =

Answer:

Given,

(6x – 24)° = 96°

Let us solve the equation

6x – 24 = 96

6x = 96 + 24

6x = 120

x = 120 ÷ 6

x = 20

D. Name a pair of adjacent angles in Figure 2.

Answer:

∠FGB and ∠BGJ are a pair of adjacent angles.

Connect to Vocabulary

The word adjacent means “next to” or “connected to.” Adjacent angles are two angles in the same plane with a common vertex and a common side, but no common interior points.

∠ __________ and ∠ __________

Answer:

∠FGB and ∠BGJ are adjacent angles.

E. Given m∠FGJ = 84°, m∠FGB = (4x)°, and m∠JGB (5x + 3)°, write an equation that can be used to determine the value of x. Solve for x.

+ = and x =

Answer:

The value of x = 9

4x = 36°

(5x + 3) = 48°

Explanation:

Given,

m∠FGJ = 84°, m∠FGB = (4x)° and m∠JGB (5x + 3)°

m∠FGB + m∠JGB = m∠FGJ

4x° + (5x + 3)° = 84°

4x + 5x + 3 = 84

9x + 3 = 84

9x = 84 – 3

9x = 81

x = 81 ÷ 9

x = 9°

4x = 4 × 9 = 36°

(5x + 3) = ((5 × 9) + 3)

= 45 + 3

= 48°

F. Explain why ∠FGJ and ∠FGB are not adjacent.

Answer:

Adjacent angles must be next to each other and they do share a common line but the given angles overlap each other. Hence ∠FGJ and ∠FGB are not adjacent.

**Turn and Talk** Explain why ∠FGB does not have a vertical angle in Figure 2.

Answer:

When two lines meet each other at a point that is called a vertical angle. The angle ∠FGB does not meet at any point. So ∠FGB does not have a vertical angle.

Question 3.

In Figure 3, ∠AOD measures (3x + 15)°.

A. What is the sum of the measures of ∠AOD and ∠DOB?

Answer:

The sum of the measures of ∠AOD and ∠DOB is 25°.

Explanation:

Given ∠AOD = (3x + 15)° ∠DOB = 90°

We need to find the sum of the measures of ∠AOD and ∠DOB.

∠AOD + ∠DOB = 180°

(3x + 15)° + 90° = 180°

(3x + 15)° = 180° – 90°

(3x + 15)° = 90°

3x = 90° – 15°

3x = 75°

x = 75 ÷ 3

x = 25°

B. If the measure of ∠DOB is (2x + 15)°, write an equation to determine the value of x.

+ =

Answer:

(2x + 15)° + (3x + 15)° = 90°

Explanation:

Given,

∠DOB = (2x + 15)°

∠AOD = (3x + 15)°

Sum of complimentary angles = 180

∠DOB + ∠AOD = 180°

(2x + 15)° + (3x + 15)° = 180°

C. Solve for x.

x = ___________

Answer:

x = 30°

Explanation:

∠DOB + ∠AOD = 180°

(2x + 15)° + (3x + 15)° = 180°

2x + 3x + 15 + 15 = 180°

5x + 30 = 180

5x = 180 – 30

5x = 150

x = 150 ÷ 5

x = 30°

D. What is the measure of ∠AOD?

Answer:

∠AOD = 105°

Explanation:

∠AOD = (3x + 15)°

∠AOD = ((3 × 30)) + 15

∠AOD = 90 + 15

∠AOD = 105°

E. What angle makes vertical angles with ∠AOD? What is its measure?

m ___________ = ____________

Answer:

∠BOC and its measure is 105°.

Explanation:

The angle that makes vertical angles with ∠AOD is ∠BOC.

F. What two angles are adjacent to ∠AOD? What are their angle measures?

m ___________ = _____________

m ___________ = ____________

Answer:

∠AOC = 75° and ∠DOB = 75°

Explanation:

Given,

∠DOB = (2x + 15)°

x = 30°

(2 × 30) + 15

60 + 15 = 75

∠DOB = 75°

**Check Understanding**

Question 1.

Two angles are complementary. The first angle measures (2x + 15)°, and the second measures (4x + 9)°. Write an equation to determine the value of x. Then solve your equation and find the measures of both angles.

Answer:

x = 11°, (2x + 15)° = 37° and (4x + 9)° = 53°

Explanation:

Given,

First angle = (2x + 15)° and second angle = (4x + 9)°

And the two angles are complementary

(2x + 15)° + (4x + 9)° = 90°

2x + 4x + 15 + 9 = 90°

6x + 24 = 90°

6x = 90 – 24

6x = 66

x = 66 ÷ 6

x = 11°

First angle = (2x + 15)°

= (2 × 11) + 15

= 22 + 15

= 37°

Second angle = (4x + 9)°

= (4 × 11) + 9

= 44 + 9

= 53°

Question 2.

∠A and ∠B are adjacent. The sum of their measures is 92°. ∠A measures (2x + 5)°. ∠B is three times the size of ∠A. Write an equation to determine the value of x. Then solve your equation and find the measures of both angles.

Answer:

x = 9°, ∠A = 23° and ∠B = 69°

Explanation:

Given,

∠A = (2x + 5)°, ∠B = 3(2x + 5)°

The sum of their measures is 92°

∠A + ∠B = 92°

(2x + 5)° + 3(2x + 5)° = 92°

(2x + 5)° + 6x + 15 = 92°

2x + 6x + 5 + 15 = 92°

8x + 20 = 92°

8x = 92° – 20

8x = 72

x = 72 ÷ 8

x = 9°

∠A = (2x + 5)°

= (2 × 9) + 5

= 18 + 5

= 23°

∠B = 3(2x + 5)°

= 6x + 15

= (6 × 9) + 15

= 54 + 15

= 69°

**On Your Own**

Question 3.

Angles A and B are adjacent angles and are supplementary. The measure of ∠A is (3x + 10)°, and the measure of ∠B is (12x + 35)°.

A. Write an equation that can be used to determine the value of x.

Answer:

∠A + ∠B = 180°

(3x + 10)° + (12x + 35)° = 180°

Explanation:

An equation that can be used to determine the value of x is (3x + 10)° + (12x + 35)° = 180°.

B. What is the value of x?

Answer:

The value of x is 9°

Explanation:

Given equation is

(3x + 10)° + (12x + 35)° = 180°

3x + 12x + 10 + 35 = 180°

15x + 45° = 180°

15x = 180° – 45°

15x = 135°

x = 135° ÷ 15

x = 9°

C. What is the measure of ∠A?

Answer:

∠A = 37°

Explanation:

Given,

∠A = (3x + 10)°

x = 9°

= (3 × 9) + 10

= 27 + 10

= 37°

Question 4.

Describe ∠COA in relation to ∠DOB

Answer:

∠COA and ∠DOB are vertically opposite angles.

∠DOB = 60°

Therefore ∠COA = ∠DOB

∠COA = 60°

Question 5.

**Attend to Precision** ∠A is complementary to ∠B. The measure of ∠A is (8x + 12)°. The measure of ∠B is half the measure of ∠A. Write an equation that can be used to determine the value of x. Then solve for x.

Answer:

x = 6

Explanation:

∠A is complimentry to ∠B

∠A + ∠B = 90°

∠A = (8x + 12)°, ∠B = (8x + 12) ÷ 2

∠B = (4x + 6)

(8x + 12) + (4x + 6) = 90°

8x + 4x + 12 + 6 = 90°

12x + 18 = 90°

12x = 90° – 18°

12x = 72°

x = 72 ÷ 12

x = 6°

**For Problems 6-7, use Figure 4.**

Question 6.

Describe the relationship between ∠BOC and ∠COD. What is the sum of their angle measures?

Answer:

∠BOC and ∠COD are adjacent angles.

The sum of their angle is ∠BOC + ∠COD = 180°

Question 7.

Describe the relationship between ∠AOB and ∠COD. Are their measures equal?

Answer:

∠AOB and ∠COD are vertically opposite angles. And their measures are equal.

Question 8.

**STEM** A rocket blasts off at a 90° angle from Earth. A second rocket launches at a different angle as shown in the diagram.

A. Write an equation that can be used to determine the value of x.

Answer:

(3x + 7) + (5x + 3) = 90°

B. What is the value of x?

Answer:

x = 10°

Explanation:

Given equation

(3x + 7) + (5x + 3) = 90°

Let us solve the given equation

3x + 5x + 7 + 3 = 90°

8x + 10 = 90°

8x = 90 – 10

8x = 80°

x = 80 ÷ 8

x = 10°

C. What is the measure of the angle of the second rocket launch in relation to Earth?

Answer:

The measure of the angle of the second rocket launch in relation to Earth is 53°

Explanation:

Given angle for second rocket is (5x + 3)

x = 10°

(5x + 3) = (5 × 10) + 3

= 50 + 3

= 53°

Question 9.

Two lines intersect to form an X. The measure of one angle is 58°.

A. What is the measure of one of the angles sharing a side with the 58° angle? Explain.

Answer:

122°

Explanation:

A straight line forms a flat angle.

180° – 58° = 122°

B. Since the angles are sharing a side, what are they called?

Answer:

As the angles are sharing a side they are called adjacent angles.

**Model with Mathematics** For Problems 10-12, write an equation that can be used to determine the value of x.

Question 10.

Answer:

2x° + 30° = 90°

Explanation:

An equation that can be used to determine the value of x is 2x° + 30° = 90°.

Question 11.

Answer:

(3x + 8)° + (x + 4)° = 180°

Explanation:

An equation that can be used to determine the value of x is (3x + 8)° + (x + 4)° = 180°.

Question 12.

Answer:

x° + x° + 100° = 180°

Explanation:

An equation that can be used to determine the value of x is x° + x° + 100° = 180°.

Question 13.

Look for Repeated Reasoning Based on the diagram, if one angle measure is given, how can you determine all of the other angle measures?

Answer:

∠A = ∠C and ∠B = ∠D

These are vertically opposite angles.

∠A + ∠B + ∠C + ∠D = 360°

**For Problems 14-15, use the expressions given for the measures of complementary angles to solve for x. Then find the angle measures.**

Question 14.

(5x + 14)° and (3x + 20)°

Answer:

Given expression

(5x + 14)° and (3x + 20)°

Let us solve the given equation

Sum of two complementary angles is 90°

5x + 14 + 3x + 20 = 90°

8x + 34 = 90°

8x = 90° -34°

8x° = 56°

x° = 56 ÷ 8

x° = 7

5x° = 5 × 7 = 35°

8x° = 8 × 7 = 56°

Question 15.

(4x + 3)° and (4x + 7)°

Answer:

x = 10°

4x = 4 × 10 = 40°

Explanation:

Let us solve the given equation

The Sum of two complementary angles is 90°

(4x + 3)° + (4x + 7)° = 90°

4x + 4x + 3 + 7 = 90°

8x + 10 = 90°

8x = 90 – 10

8x = 80°

x = 80 ÷ 8

x = 10°

4x = 4 × 10 = 40°

**For Problems 16-17, use the expressions given for the measures of supplementary angles to solve for x. Then find the angle measures.**

Question 16.

(9x + 17)° and (6x + 13)°

Answer:

x = 10°

9x = 9 × 10 = 90°

6x = 6 × 10 = 60°

Explanation:

Given equation is

(9x + 17)° and (6x + 13)°

The sum of two supplementary angles is 180°

9x + 17 + 6x + 13 = 180°

15x + 30 = 180

15x = 180 – 30

15x = 150

x = 150 ÷ 15

x = 10°

9x = 9 × 10 = 90°

6x = 6 × 10 = 60°

Question 17.

(6x + 7)° and (5x + 8)°

Answer:

x = 15°

6x = 6 × 15 = 90°

5x = 5 × 15 = 75°

Explanation:

Given (6x + 7)° and (5x + 8)°

The sum of two supplementary angles is 180°

(6x + 7)° + (5x + 8)° = 180°

6x + 5x + 7 + 8 = 180°

11x + 15 = 180°

11x = 180° – 15°

11x = 165°

x = 165° ÷ 11

x = 15°

6x = 6 × 15 = 90°

5x = 5 × 15 = 75°

Question 18.

An angle measures (2x + 11)°.

A. What is the measure of an angle that is vertical to the given angle?

Answer:

(79 – 2x)°

Explanation:

90 – (2x + 11)° =

90 – 11 = 2x

2x = 79

(79 – 2x)°

B. Write an expression to represent the measure of an angle supplementary to the given angle.

Answer:

(169 – 2x)°

Explanation:

180° – (2x + 11)°

= (169 – 2x)°

Question 19.

The diagram shows a right angle. What does x equal? What are the angle measures?

Answer:

x = 16°, (3x + 5)° = 53, (2x + 5)° = 37°

Explanation:

Sum of angles = 90°

(3x + 5)° + (2x + 5)° = 90°

5x + 10 = 90

5x = 90° -10°

5x = 80°

x = 80°÷5

x = 16°

(3x + 5)° = ((3 × 16) + 5)°

(48 + 5)°

53°

(2x + 5)° = ((2 × 16) + 5)°

= 32 + 5

= 37°

**Lesson 7.5 More Practice/Homework**

Question 1.

Libby is putting together a piece of furniture. She notices that two of the pieces form a right angle. If these right angles are cut in half, or bisected, by another bar, what would each angle measure within that right angle?

Answer:

Each angle within that right angle measures 45 degrees.

Explanation:

The given pieces of furniture form Right angle.

We need to find each angle measure within that right angle

The furniture is attached such that the angle between them is a right angle.

The right angle measures 90°.

Another bar bisects the right angle.

Bisecting the right angle means it is divided into two halves.

So \(\frac{90}{2}\) = 45°

So each angle within that right angle measures 45 degrees.

Question 2.

The diagram shows a right angle. Write an equation to determine the value of x. Solve for x.

Answer:

x = 6°, 8x° = 48, (6x + 6)° = 40°

Explanation:

The given diagram is a right angle triangle that has 90°

8x° + (6x + 6)° = 90°

8x° + 6x + 6 = 90°

14x + 6 = 90°

14x = 90 – 6

14x = 84

x = 84 ÷ 14

x = 6°

8x = 6 × 8 = 48

(6x + 6)° = ((6 × 6) + 6)°

= 36 + 6

= 40°

Question 3.

A given angle measures 30°, and the measure of its vertical angle is expressed as (5x + 5)°.

A. Write an equation to determine the value of x. Solve for x.

Answer:

The required equation is 5x + 5 = 30

5x + 5 = 30

x = 5

B. If the measure of an angle adjacent to the given angle is represented by the expression (24x + 30)° using the same value for x, what is the measure of the adjacent angle?

Answer:

Vertically opposite angles are equal, then

5x + 5 = 30

5x = 30 – 5

5x = 25

x = 25 ÷ 5

x = 5

The required equation is 5x + 5 = 30

Question 4.

**Attend to Precision** Ms. Baumgartner draws a pair of supplementary angles and tells the class that the angle measures are (4x + 30)° and (2x + 6)°.

A. Write an equation to determine the value of x. Solve for x.

Answer:

(4x + 30)° + (2x + 6)° = 180°

x = 24

Explanation:

An equation to determine the value of x is (4x + 30)° + (2x + 6)° = 180°

Now let us solve the given equation

4x + 2x + 30 + 6 = 180°

6x + 36 = 180°

6x = 180° – 36°

6x = 144°

x = 144° ÷ 6°

x = 24°

4x + 30 = ((4 × 24) + 30)

= 96 + 30

= 126°

2x + 6 = ((2 × 24) + 6)

= (48 + 6)

= 54°`

B. What does the larger angle measure? What does the smaller angle measure?

Answer:

The larger angle measure is 126°. The smaller angle measures 54°.

Question 5.

**Math on the Spot** Use the diagram to find m∠2 if m∠1 = 105°.

Answer:

m∠2 = 75°

Explanation:

Given m∠1 = 105°

sum of supplementary angles = 180°

m∠1 + m∠2 = 180°

105° + m∠2 = 180°

m∠2 = 180° – 105°

m∠2 = 75°

**Test Prep**

Question 6.

Consider adjacent angles that measure (2x + 45)° and (3x + 55)°. The sum of the measures of these two angles is 135°.

A. Write and solve an equation to find the value of x.

Answer:

The equation to find the value of x is (2x + 45)° + (3x + 55)° = 135°.

Value of x = 11°

Explanation:

Given angles are (2x + 45)° and (3x + 55)°

sum of the measures of these two angles is 135°

(2x + 45)° + (3x + 55)° = 135°

2x + 3x + 45 + 35 = 135°

5x + 80° = 135°

5x = 135° – 80°

5x = 55°

x = 55 ÷ 5

x = 11°

B. Using the value of x, what is the angle measure represented by the expression (2x + 45)°?

Answer:

The angle measure represented by the expression (2x + 45)° is 67°.

Explanation:

Given x = 11°

Angle given is (2x + 45)°

(2 × 11) + 45

22 + 45

67°

Question 7.

An angle has a measure of (3x + 5)°, and its complementary angle has a measure of (2x + 5)°. Which is the correct equation to find x?

(A) 5x + 10 = 180

(B) 5x = 180 + 10

(C) 5x + 10 = 90

(D) 5x = 90 + 10

Answer:

The correct equation to find x is 5x + 10 = 90°.

Explanation:

Given angles are (3x + 5)° and (2x + 5)°

Sum of complementary angles are 90°

3x + 5 + 2x + 5 = 90°

5x + 10 = 90°

Question 8.

Vertical angles have the same measure. True or False?

Answer:

True.

Explanation:

vertical angles are always in pairs. They have a common vertex but they cannot share a side. Vertical angles are congruent, which means they have equal measure.

Question 9.

The sum of the measures of adjacent angles is always 90°. True or False?

Answer:

False.

Explanation:

The sum of the measures of adjacent angles is 180°.

Question 10.

Draw lines to match.

Answer:

**Spiral Review**

Question 11.

Frankie and Marcel are picking apples. Frankie has 18 apples, which is 4 times plus 2 more apples than Marcel has. How many apples does Marcel have?

Answer:

Marcel has 4 apples.

Explanation:

Frankie has 18 apples that are 4 times more and 2 more apples of what marcel has then the equation is

(18 – 2) ÷ 4

16 ÷ 4

= 4

Question 12.

Is 10 a solution of the inequality x ≥ 12?

Answer:

No

Explanation:

Given inequality x ≥ 12

If x = 10

10 is not greater than 12

10 ≥ 12 The inequality is wrong.