## 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:

→ Draw a point B on the minor arc between A and C.

Students draw point B.

→ Draw the arc intercepted by ∠ABC. Make it red in your diagram.

Students draw the arc and color it red.

→ 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.

→ Using your protractor, measure ∠ABC. Write your answer on your diagram.

Answers will vary.

Now measure the arc in degrees.

→ We could measure ∠AOC and then subtract that measure from 360°.

→ 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 ∠GBF is 17°, name three other angles that have the same measure and explain why.

Answer:

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

What is the measure of ∠GAF? Explain.

Answer:

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

Can you find the measure of ∠BAD? Explain.

Answer:

34°; ∠BAD and ∠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. m∠BAC

Answer:

36°

b. m∠DAE

Answer:

108°

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

Answer:144°

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

Answer:

180°

Exercise 2.

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

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

Answer:

60°

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

Answer:

300°

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

Answer:

180°

Exercise 3.

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

a. m∠BAE

Answer:

20°

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

Answer:

160°

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

Answer:

260°

Exercise 4.

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

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

Answer:

323°

b. m∠CBD

Answer:

18.5°

c. m∠CED

Answer:

161.5°

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

Question 1.

Given circle A with m∠CAD = 50°,

a. Name a central angle.

Answer:

∠CAD

b. Name an inscribed angle.

Answer:

∠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:

50°

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

Answer:

310°

h. Find m∠CBD.

Answer:

25°

Question 2.

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

Answer:

\(m \widehat{B E}\) = 64°

\(m \widehat{C D}\) = 64°

\(m \widehat{C E}\) = 116°

\(m \widehat{B D}\) = 116°

Question 3.

Given circle A, find the following measure.

a. m∠BAD

Answer:

100°

b. m∠CAB

Answer:

80°

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

Answer:

80°

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

Answer:

100°

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

Answer:

260°

Question 4.

Find the measure of angle x.

Answer:

33°

Question 5.

In the figure, m∠BAC = 126° and m∠BED = 32°. Find m∠DEC.

Answer:

85°

Question 6.

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

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

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

m∠KDC = \(\frac{42^{\circ}}{2}\) = 21˚

a = _________________________________

In △BCD, b = _________________________________

c = _________________________________

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

m∠JCD = _________________________________

d = _______________ _________________________________

m∠KBD = a + b = _________________________________

m∠CKJ = c + d = _________________________________

Answer:

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

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

a =

21°

Congruent angles inscribed in same arc

In △BCD, b = 64° Sum of angles of triangle is 180°

c = 64° Congruent angles inscribed in same arc

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

m∠JCD = 37° Angle bisector

d = 37° Congruent angles inscribed in same arc

m∠KBD = a + b = 85°

m∠CKJ = c + d = 101°

### 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 \(m \widehat{C D}\) = 56°.

a. Name a central angle.

Answer:

∠CAD

b. Name an inscribed angle.

Answer:

Answers will vary. ∠CED

c. Name a chord that is not a diameter.

Answer:

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

d. What is the measure of ∠CAD?

Answer:

56°

e. What is the measure of ∠CBD?

Answer:

28°

f. Name 3 angles of equal measure.

Answer:

m∠CED = m∠CFD = m∠CBD

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

Answer:

180°