## Engage NY Eureka Math Geometry Module 5 Lesson 7 Answer Key

### Eureka Math Geometry Module 5 Lesson 7 Example Answer Key

Example

What if we started with an angle inscribed in the minor arc between A and C?

Answer:

â†’ Draw a point B on the minor arc between A and C.

Students draw point B.

â†’ Draw the arc intercepted by âˆ ABC. Make it red in your diagram.

Students draw the arc and color it red.

â†’ In your diagram, do you think the measure of an arc between A and C is half of the measure of the inscribed angle? Why or why not?

The phrasing and explanations can vary. However, there is one answer; the measure of the inscribed arc is twice the measure of the inscribed angle.

â†’ Using your protractor, measure âˆ ABC. Write your answer on your diagram.

Answers will vary.

Now measure the arc in degrees.

â†’ We could measure âˆ AOC and then subtract that measure from 360Â°.

â†’ Yes, the measure of the inscribed angle is half the measure of its intercepted arc.

### Eureka Math Geometry Module 5 Lesson 7 Exercise Answer Key

Opening Exercise

If the measure of âˆ GBF is 17Â°, name three other angles that have the same measure and explain why.

Answer:

Answers will vary. âˆ GHF, âˆ GCF, âˆ GDF, âˆ GEF all have the same measure because they are inscribed in the same arc.

What is the measure of âˆ GAF? Explain.

Answer:

34Â°; it is the central angle with an inscribed arc of 17Â°. The measure of the central angle is double the measure of the inscribed angle of the same arc.

Can you find the measure of âˆ BAD? Explain.

Answer:

34Â°; âˆ BAD and âˆ GAF are vertical angles and are congruent.

Exercises

Exercise 1.

In circle A, \(m \widehat{B C}\) : \(m \widehat{C E}\) : \(m \widehat{E D}\) : \(m \widehat{B D}\) = 1 : 2 : 3 : 4. Find the following angles of measure.

a. mâˆ BAC

Answer:

36Â°

b. mâˆ DAE

Answer:

108Â°

c. \(m \widehat{B D}\)

Answer:144Â°

d. \(m \widehat{C E D}\)

Answer:

180Â°

Exercise 2.

In circle B, AB = CD. Find the following angles of measure.

a. \(m \widehat{C D}\)

Answer:

60Â°

b. \(m \widehat{C A D}\)

Answer:

300Â°

c. \(m \widehat{A C D}\)

Answer:

180Â°

Exercise 3.

In circle A, \(\overline{B C}\) is a diameter and mâˆ DAC = 100Â°. If \(m \widehat{E C}\) = 2\(m \widehat{B D}\), find the following angles of measure.

a. mâˆ BAE

Answer:

20Â°

b. \(m \widehat{E C}\)

Answer:

160Â°

c. \(m \widehat{D E C}\)

Answer:

260Â°

Exercise 4.

Given circle A with mâˆ CAD = 37Â°, find the following angles of measure.

a. \(m \widehat{C B D}\)

Answer:

323Â°

b. mâˆ CBD

Answer:

18.5Â°

c. mâˆ CED

Answer:

161.5Â°

### Eureka Math Geometry Module 5 Lesson 7 Problem Set Answer Key

Question 1.

Given circle A with mâˆ CAD = 50Â°,

a. Name a central angle.

Answer:

âˆ CAD

b. Name an inscribed angle.

Answer:

âˆ CBD

c. Name a chord.

Answer:

Answers will vary. \(\overline{B D}\)

d. Name a minor arc.

Answer:

Answers will vary. \(\widehat{C D}\)

e. Name a major arc.

Answer:

\(\widehat{C B D}\)

f. Find \(m \widehat{C D}\)

Answer:

50Â°

g. Find \(m \widehat{C B D}\).

Answer:

310Â°

h. Find mâˆ CBD.

Answer:

25Â°

Question 2.

Given circle A, find the measure of each minor arc.

Answer:

\(m \widehat{B E}\) = 64Â°

\(m \widehat{C D}\) = 64Â°

\(m \widehat{C E}\) = 116Â°

\(m \widehat{B D}\) = 116Â°

Question 3.

Given circle A, find the following measure.

a. mâˆ BAD

Answer:

100Â°

b. mâˆ CAB

Answer:

80Â°

c. \(m \widehat{B C}\)

Answer:

80Â°

d. \(m \widehat{B D}\)

Answer:

100Â°

e. \(m \widehat{B C D}\)

Answer:

260Â°

Question 4.

Find the measure of angle x.

Answer:

33Â°

Question 5.

In the figure, mâˆ BAC = 126Â° and mâˆ BED = 32Â°. Find mâˆ DEC.

Answer:

85Â°

Question 6.

In the figure, mâˆ BCD = 74Â° and mâˆ BDC = 42Â°. K is the midpoint of \(\widehat{C B}\), and J is the midpoint of \(\widehat{B D}\). Find mâˆ KBD and mâˆ CKJ.

Solution: Join BK, KC, KD, KJ, JC, and JD.

\(m \widehat{B K}\) = \(m \widehat{K C}\)

mâˆ KDC = \(\frac{42^{\circ}}{2}\) = 21Ëš

a = _________________________________

In â–³BCD, b = _________________________________

c = _________________________________

\(m \widehat{B J}\) = \(m \widehat{J D}\)

mâˆ JCD = _________________________________

d = _______________Â Â Â _________________________________

mâˆ KBD = a + b = _________________________________

mâˆ CKJ = c + d = _________________________________

Answer:

\(m \widehat{B K}\) = \(m \widehat{K C}\) Midpoint forms arcs of equal measure

mâˆ KDC = \(\frac{42^{\circ}}{2}\) = 21Â° Angle bisector

a =

21Â°

Congruent angles inscribed in same arc

In â–³BCD, b = 64Â° Sum of angles of triangle is 180Â°

c = 64Â° Congruent angles inscribed in same arc

\(m \widehat{B J}\) = \(m \widehat{J D}\) Midpoint forms arcs of equal measure

mâˆ JCD = 37Â° Angle bisector

d = 37Â°Â Â Â Â Congruent angles inscribed in same arc

mâˆ KBD = a + b = 85Â°

mâˆ CKJ = c + d = 101Â°

### Eureka Math Geometry Module 5 Lesson 7 Exit Ticket Answer Key

Question 1.

Given circle A with diameters \(\overline{B C}\) and \(\overline{D E}\) and \(m \widehat{C D}\) = 56Â°.

a. Name a central angle.

Answer:

âˆ CAD

b. Name an inscribed angle.

Answer:

Answers will vary. âˆ CED

c. Name a chord that is not a diameter.

Answer:

Answers will vary. \(\overline{C E}\)

d. What is the measure of âˆ CAD?

Answer:

56Â°

e. What is the measure of âˆ CBD?

Answer:

28Â°

f. Name 3 angles of equal measure.

Answer:

mâˆ CED = mâˆ CFD = mâˆ CBD

g. What is the degree measure of \(\widehat{C D B}\)?

Answer:

180Â°