# Eureka Math Grade 7 Module 3 Lesson 10 Answer Key

## Engage NY Eureka Math 7th Grade Module 3 Lesson 10 Answer Key

### Eureka Math Grade 7 Module 3 Lesson 10 Example Answer Key

Example 1.
Estimate the measurement of x. ___
In a complete sentence, describe the angle relationship in the diagram.

âˆ BAC and âˆ CAD are angles on a line and their measures have a sum of 180Â°.

Write an equation for the angle relationship shown in the figure and solve for x. Then, find the measures of âˆ BAC and confirm your answers by measuring the angle with a protractor.
x + 132 = 180
x + 132 – 132 = 180 – 132
x = 48
mâˆ BAC = 48Â°

Example 2.
In a complete sentence, describe the angle relationship in the diagram.

âˆ AEL and âˆ LEB are angles on a line and their measures have a sum of 180Â°. âˆ AEL and âˆ KEB are vertical angles and are of equal measurement.

Write an equation for the angle relationship shown in the figure and solve for x and y. Find the measurements of âˆ LEB and âˆ KEB.
y = 144Â°; mâˆ KEB = 144Â° (or vert. âˆ s are = )
x + 144 = 180
x + 144 – 144 = 180 – 144
x = 36
mâˆ LEB = 36Â°

Example 3.
In a complete sentence, describe the angle relationships in the diagram.

âˆ GKE, âˆ EKF, and âˆ GKF are angles at a point and their measures have a sum of 360Â°.

Write an equation for the angle relationship shown in the figure and solve for x. Find the measurement of âˆ EKF and confirm your answers by measuring the angle with a protractor.
x + 90 + 135 = 360
x + 225 = 360
x + 225 – 225 = 360 – 225
x = 135
mâˆ EKF = 135Â°

Example 4.
The following two lines intersect. The ratio of the measurements of the obtuse angle to the acute angle in any adjacent angle pair in this figure is 2:1. In a complete sentence, describe the angle relationships in the diagram.

The measurement of an obtuse angle is twice the measurement of an acute angle in the diagram.

Label the diagram with expressions that describe this relationship. Write an equation that models the angle relationship and solve for x. Find the measurements of the acute and obtuse angles.
2x + 1x = 180
3x = 180
($$\frac{1}{3}$$)(3x) = ($$\frac{1}{3}$$)(180)
x = 60
Acute angle = 60Â°
Obtuse angle = 2xÂ° = 2(60Â°) = 120Â°

### Eureka Math Grade 7 Module 3 Lesson 10 Opening Exercise Answer Key

Use the diagram to complete the chart.

### Eureka Math Grade 7 Module 3 Lesson 10 Exercise Answer Key

Exercise 1.
In a complete sentence, describe the angle relationship in the diagram.

âˆ BAC, âˆ CAD, and âˆ DAE are angles on a line and their measures have a sum of 180Â°.

Find the measurements of âˆ BAC and âˆ DAE.
3x + 90 + 2x = 180
5x + 90 = 180
5x + 90 – 90 = 180 – 90
($$\frac{1}{5}$$)(5x) = ($$\frac{1}{5}$$)(90)
x = 18
mâˆ BAC = 3(18Â°) = 54Â°
mâˆ DAE = 2(18Â°) = 36Â°

Exercise 2.
In a complete sentence, describe the angle relationships in the diagram.

âˆ JEN and âˆ NEM are adjacent angles and, when added together, are the measure of âˆ JEM; âˆ JEM and âˆ KEL are vertical angles and are of equal measurement.

Write an equation for the angle relationship shown in the figure and solve for x.
3x + 16 = 85
3x + 16 – 16 = 85 – 16
3x = 69
($$\frac{1}{3}$$)3x = 69($$\frac{1}{3}$$)
x = 23

Exercise 3.
In a complete sentence, describe the angle relationships in the diagram.

âˆ EAH, âˆ GAH, âˆ GAF, and âˆ FAE are angles at a point and their measures sum to 360Â°.

Find the measurement of âˆ GAH.
(x + 1) + 59 + 103 + 167 = 360
x + 1 + 59 + 103 + 167 = 360
x = 30
mâˆ GAH = (30 + 1)Â° = 31Â°

Exercise 4.
The ratio of mâˆ GFH to mâˆ EFH is 2âˆ¶3. In a complete sentence, describe the angle relationships in the diagram.

The measurement of âˆ GFH is $$\frac{2}{3}$$ the measurement of âˆ EFH; The measurements of âˆ GFH and âˆ EFH have a sum of 90Â°.

Find the measures of âˆ GFH and âˆ EFH.
2x + 3x = 90
5x = 90
($$\frac{1}{5}$$)(5x) = ($$\frac{1}{5}$$)(90)
x = 18
mâˆ GFH = 2(18Â°) = 36Â°
mâˆ EFH = 3(18Â°) = 54Â°

### Eureka Math Grade 7 Module 3 Lesson 10 Problem Set Answer Key

For each question, use angle relationships to write an equation in order to solve for each variable. Determine the indicated angles. You can check your answers by measuring each angle with a protractor.

Question 1.
In a complete sentence, describe the relevant angle relationships in the following diagram. Find the measurement of âˆ DAE.

One possible response: âˆ CAD, âˆ DAE, and âˆ FAE are angles on a line and their measures sum to 180Â°.
90 + x + 65 = 180
x + 155 = 180
x + 155 – 155 = 180 – 155
x = 25
mâˆ DAE = 25Â°

Question 2.
In a complete sentence, describe the relevant angle relationships in the following diagram. Find the measurement of âˆ QPR.

âˆ QPR, âˆ RPS, and âˆ SPT are angles on a line and their measures sum to 180Â°.
f + 154 + f = 180
2f + 154 = 180
2f + 154 – 154 = 180 – 154
2f = 26
($$\frac{1}{2}$$)2f = ($$\frac{1}{2}$$)26
f = 13
mâˆ QPR = 13Â°

Question 3.
In a complete sentence, describe the relevant angle relationships in the following diagram. Find the measurements of âˆ CQD and âˆ EQF.

âˆ BQC, âˆ CQD, âˆ DQE, âˆ EQF, and âˆ FQG are angles on a line and their measures sum to 180Â°.
10 + 2x + 103 + 3x + 12 = 180
5x + 125 = 180
5x + 125 – 125 = 180 – 125
5x = 55
($$\frac{1}{5}$$)5x = ($$\frac{1}{5}$$)55
x = 11
mâˆ CQD = 2(11Â°) = 22Â°
mâˆ EQF = 3(11Â°) = 33Â°

Question 4.
In a complete sentence, describe the relevant angle relationships in the following diagram. Find the measure of x.

All of the angles in the diagram are angles at a point and their measures sum to 360Â°.
4(x + 71) = 360
4x + 284 = 360
4x + 284 – 284 = 360 – 284
4x = 76
($$\frac{1}{4}$$)4x = ($$\frac{1}{4}$$)76
x = 19

Question 5.
In a complete sentence, describe the relevant angle relationships in the following diagram. Find the measures of x and y.

âˆ CKE, âˆ EKD, and âˆ DKB are angles on a line and their measures sum to 180Â°. Since âˆ FKA and âˆ AKE form a straight angle and the measurement of âˆ FKA is 90Â°, âˆ AKE is 90Â°, making âˆ CKE and âˆ AKC form a right angle and their measures have a sum of 90Â°.
x + 25 + 90 = 180
x + 115 = 180
x + 115 – 115 = 180 – 115
x = 65

(65) + y = 90
65 – 65 + y = 90 – 65
y = 25

Question 6.
In a complete sentence, describe the relevant angle relationships in the following diagram. Find the measures of x and y.

âˆ EAG and âˆ FAK are vertical angles and are of equal measurement. âˆ EAG and âˆ GAD form a right angle and their measures have a sum of 90Â°.
2x + 24 = 90
2x + 24 – 24 = 90 – 24
2x = 66
($$\frac{1}{2}$$)2x = ($$\frac{1}{2}$$)66
x = 33
3y = 66
($$\frac{1}{3}$$)3y = ($$\frac{1}{3}$$)66
y = 22

Question 7.
In a complete sentence, describe the relevant angle relationships in the following diagram. Find the measures of âˆ CAD and âˆ DAE.

âˆ CAD and âˆ DAE form a right angle and their measures have a sum of 90Â°.
($$\frac{3}{2}$$ x + 20) + 2x = 90
$$\frac{7}{2}$$ x + 20 = 90
$$\frac{7}{2}$$ x + 20 – 20 = 90 – 20
$$\frac{7}{2}$$ x = 70
($$\frac{2}{7}$$) $$\frac{7}{2}$$ x = 70($$\frac{2}{7}$$)
x = 20
mâˆ CAD = $$\frac{3}{2}$$ (20Â°) + 20Â° = 50Â°
mâˆ DAE = 2(20Â°) = 40Â°

Question 8.
In a complete sentence, describe the relevant angle relationships in the following diagram. Find the measure of âˆ CQG.

âˆ DQE and âˆ CQF are vertical angles and are of equal measurement. âˆ CQG and âˆ GQF are adjacent angles and their measures sum to the measure of âˆ CQF.
3x + 56 = 155
3x + 56 – 56 = 155 – 56
3x = 99
($$\frac{1}{3}$$)3x = ($$\frac{1}{3}$$)99
x = 33
mâˆ CQG = 3(33Â°) = 99Â°

Question 9.
The ratio of the measures of a pair of adjacent angles on a line is 4:5.
a. Find the measures of the two angles.
âˆ 1 = 4x, âˆ 2 = 5x
4x + 5x = 180
9x = 180
($$\frac{1}{9}$$)9x = ($$\frac{1}{9}$$)180
x = 20
âˆ 1 = 4(20Â°) = 80Â°
âˆ 2 = 5(20Â°) = 100Â°

b. Draw a diagram to scale of these adjacent angles. Indicate the measurements of each angle.

Question 10.
The ratio of the measures of three adjacent angles on a line is 3:4:5.
a. Find the measures of the three angles.
âˆ 1 = 3x, âˆ 2 = 4x, âˆ 3 = 5x
3x + 4x + 5x = 180
12x = 180
($$\frac{1}{12}$$)12x = ($$\frac{1}{12}$$)180
x = 15
âˆ 1 = 3(15Â°) = 45Â°
âˆ 2 = 4(15Â°) = 60Â°
âˆ 3 = 5(15Â°) = 75Â°

b. Draw a diagram to scale of these adjacent angles. Indicate the measurements of each angle.

### Eureka Math Grade 7 Module 3 Lesson 10 Exit Ticket Answer Key

In a complete sentence, describe the relevant angle relationships in the following diagram. That is, describe the angle relationships you could use to determine the value of x.
âˆ KAE and âˆ EAF are adjacent angles whose measurements are equal to âˆ KAF; âˆ KAF and âˆ JAG are vertical angles and are of equal measurement.

Use the angle relationships described above to write an equation to solve for x. Then, determine the measurements of âˆ JAH and âˆ HAG.

($$\frac{1}{8}$$)(8x) = ($$\frac{1}{8}$$)(120)