## Engage NY Eureka Math 8th Grade Module 2 Lesson 14 Answer Key

### Eureka Math Grade 8 Module 2 Lesson 14 Example Answer Key

Example 1

Find the measure of angle x.

Answer:

We need to find the sum of the measures of the remote interior angles to find the measure of the exterior angle x:

14+30=44. Therefore, the measure of âˆ x is 44Â°.

â†’ Present an informal argument that proves you are correct.

â†’ We know that triangles have a sum of interior angles that is equal to 180Â°. We also know that straight angles are 180Â°. âˆ ABC must be 136Â°, which means that âˆ x is 44Â°.

Example 2.

Find the measure of angle x.

Answer:

We need to find the sum of the measures of the remote interior angles to find the measure of the exterior angle x: 44+32=76. Therefore, the measure of âˆ x is 76Â°.

â†’ Present an informal argument that proves you are correct.

â†’ We know that triangles have a sum of interior angles that is equal to 180Â°. We also know that straight angles are 180Â°. âˆ ACB must be 104Â°, which means that âˆ x is 76Â°.

Example 3.

Find the measure of angle x.

Answer:

180-121=59. Therefore, the measure of âˆ x is 59Â°.

Students should notice that they are not given the two remote interior angles associated with the exterior angle x.

For that reason, they must use what they know about straight angles (or supplementary angles) to find the measure of

angle x.

Example 4.

Find the measure of angle x.

Answer:

129-45=84. Therefore, the measure of âˆ x is 84Â°.

### Eureka Math Grade 8 Module 2 Lesson 14 Exercise Answer Key

Exercises 1â€“4

Use the diagram below to complete Exercises 1â€“4.

Exercise 1.

Name an exterior angle and the related remote interior angles.

Answer:

The exterior angle is âˆ ZYP, and the related remote interior angles are âˆ YZX and âˆ ZXY.

Exercise 2.

Name a second exterior angle and the related remote interior angles.

Answer:

The exterior angle is âˆ XZQ, and the related remote interior angles are âˆ ZYX and âˆ ZXY.

Exercise 3.

Name a third exterior angle and the related remote interior angles.

Answer:

The exterior angle is âˆ RXY, and the related remote interior angles are âˆ ZYX and âˆ XZY.

Exercise 4.

Show that the measure of an exterior angle is equal to the sum of the measures of the related remote interior angles.

Answer:

Triangle XYZ has interior angles âˆ XYZ, âˆ YZX, and âˆ ZXY. The sum of those angles is 180Â°. The straight angle âˆ XYP also has a measure of 180Â° and is made up of angles âˆ XYZ and âˆ ZYP. Since the triangle and the straight angle have the same number of degrees, we can write the sum of their respective angles as an equality:

âˆ XYZ+âˆ YZX+âˆ ZXY=âˆ XYZ+ZYP.

Both the triangle and the straight angle share âˆ XYZ. We can subtract the measure of that angle from the triangle and the straight angle. Then, we have

âˆ YZX+âˆ ZXY=âˆ ZYP,

where the angle âˆ ZYP is the exterior angle, and the angles âˆ YZX and âˆ ZXY are the related remote interior angles of the triangle. Therefore, the sum of the measures of the remote interior angles of a triangle are equal to the measure of the exterior angle.

Exercise 5â€“10

Question 5.

Find the measure of angle x. Present an informal argument showing that your answer is correct.

Answer:

Since 89+28 = 117, the measure of angle x is 117Â°. We know that triangles have a sum of interior angles that is equal to 180Â°. We also know that straight angles are 180Â°. âˆ ACB must be 63Â°, which means that âˆ x is 117Â°.

Question 6.

Find the measure of angle x. Present an informal argument showing that your answer is correct.

Answer:

Since 59+52=111, the measure of angle x is 111Â°. We know that triangles have a sum of interior angles that is equal to 180Â°. We also know that straight angles are 180Â°. âˆ CAB must be 69Â°, which means that âˆ x is 111Â°.

Question 7.

Find the measure of angle x. Present an informal argument showing that your answer is correct.

Answer:

Since 180-79=101, the measure of angle x is 101Â°. We know that straight angles are 180Â°, and the straight angle in the diagram is made up of âˆ ABC and âˆ x. âˆ ABC is 79Â°, which means that âˆ x is 101Â°.

Question 8.

Find the measure of angle x. Present an informal argument showing that your answer is correct.

Answer:

Since 71+74=145, the measure of angle x is 145Â°. We know that triangles have a sum of interior angles that is equal to 180Â°. We also know that straight angles are 180Â°. âˆ ACB must be 35Â°, which means that âˆ x is 145Â°.

Question 9.

Find the measure of angle x. Present an informal argument showing that your answer is correct.

Answer:

Since 107+32=139, the measure of angle x is 139Â°. We know that triangles have a sum of interior angles that is equal to 180Â°. We also know that straight angles are 180Â°. âˆ CBA must be 41Â°, which means that x is 139Â°.

Question 10.

Find the measure of angle x. Present an informal argument showing that your answer is correct.

Answer:

Since 156-81 = 75, the measure of angle x is 75Â°. We know that triangles have a sum of interior angles that is equal to 180Â°. We also know that straight angles are 180Â°. âˆ BAC must be 24Â° because it is part of the straight angle. Then, âˆ x=180Â°-(81Â°+24Â°), which means âˆ x is 75Â°.

### Eureka Math Grade 8 Module 2 Lesson 14 Exit Ticket Answer Key

Question 1.

Find the measure of angle p. Present an informal argument showing that your answer is correct.

Answer:

The measure of angle p is 67Â°. We know that triangles have a sum of interior angles that is equal to 180Â°. We also know that straight angles are 180Â°. âˆ BAC must be 113Â°, which means that âˆ p is 67Â°.

Question 2.

Find the measure of angle q. Present an informal argument showing that your answer is correct.

Answer:

The measure of angle q is 27Â°. We know that triangles have a sum of interior angles that is equal to 180Â°. We also know that straight angles are 180Â°. âˆ CAB must be 25Â°, which means that âˆ q is 27Â°.

Question 3.

Find the measure of angle r. Present an informal argument showing that your answer is correct.

Answer:

The measure of angle r is 121Â°. We know that triangles have a sum of interior angles that is equal to 180Â°. We also know that straight angles are 180Â°. âˆ BCA must be 59Â°, which means that âˆ r is 121Â°.

### Eureka Math Grade 8 Module 2 Lesson 14 Problem Set Answer Key

Students practice finding missing angle measures of triangles.

For each of the problems below, use the diagram to find the missing angle measure. Show your work.

Question 1.

Find the measure of angle x. Present an informal argument showing that your answer is correct.

Answer:

Since 26+13=39, the measure of angle x is 39Â°. We know that triangles have a sum of interior angles that is equal to 180Â°. We also know that straight angles are 180Â°. âˆ BCA must be 141Â°, which means that âˆ x is 39Â°.

Question 2.

Find the measure of angle x.

Answer:

Since 52+44=96, the measure of angle x is 96Â°.

Question 3.

Find the measure of angle x. Present an informal argument showing that your answer is correct.

Answer:

Since 76-25=51, the measure of âˆ x is 51Â°. We know that triangles have a sum of interior angles that is equal to 180Â°. We also know that straight angles are 180Â°. âˆ BAC must be 104Â° because it is part of the straight angle. Then, x=180Â°-(104Â°+25Â°), which means âˆ x is 51Â°.

Question 4.

Find the measure of angle x.

Answer:

Since 27+52 =79, the measure of angle x is 79Â°.

Question 5.

Find the measure of angle x.

Answer:

Since 180-104=76, the measure of angle x is 76Â°.

Question 6.

Find the measure of angle x.

Answer:

Since 52+53=105, the measure of angle x is 105Â°.

Question 7.

Find the measure of angle x.

Answer:

Since 48+83=131, the measure of angle x is 131Â°.

Question 8.

Find the measure of angle x.

Answer:

Since 100+26=126, the measure of angle x is 126Â°.

Question 9.

Find the measure of angle x.

Answer:

Since 126-47=79, the measure of angle x is 79Â°.

Question 10.

Write an equation that would allow you to find the measure of angle x. Present an informal argument showing that your answer is correct.

Answer:

Since y+z=x, the measure of angle x is (y+z)Â°. We know that triangles have a sum of interior angles that is equal to 180Â°. We also know that straight angles are 180Â°.

Then, âˆ y+âˆ z+âˆ BAC=180Â°, and âˆ x+âˆ BAC=180Â°. Since both equations are equal to 180Â°,

then âˆ y+âˆ z+âˆ BAC=âˆ x+âˆ BAC. Subtract âˆ BAC from each side of the equation, and you get âˆ y+âˆ z=âˆ x.