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

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 ∠ ____________
∠CBE and ∠CBA

Explanation:
The right angles are ∠CBE and ∠CBA.

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

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

C. What is the measure of ∠ABE? ____________
∠180°

Explanation:
The measure of ∠ABE is ∠180°

D. Name a pair of supplementary angles.
∠____________ and ∠ _____________
∠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 =
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 =
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?
∠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 ∠ ____________
∠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.
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 =
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.
∠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 ∠ __________
∠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 =
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.
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.
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?
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.
+ =
(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 = ___________
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?
∠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 ___________ = ____________
∠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 ___________ = ____________
∠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.
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.
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°

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.
∠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?
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?
∠A = 37°

Explanation:
Given,
∠A = (3x + 10)°
x = 9°
= (3 × 9) + 10
= 27 + 10
= 37°

Question 4.
Describe ∠COA in relation to ∠DOB

∠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.
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?
∠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?
∠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.
(3x + 7) + (5x + 3) = 90°

B. What is the value of x?
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?
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.
122°

Explanation:
A straight line forms a flat angle.
180° – 58° = 122°

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

2x° + 30° = 90°

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

Question 11.

(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.

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?

∠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)°
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)°
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)°
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)°
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?
(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.
(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?

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?

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.

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.
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?
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.
(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?
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°.

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.
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)°?
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
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?
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?
False.

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

Question 10.
Draw lines to match.

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?
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?