# Eureka Math Geometry Module 5 Lesson 7 Answer Key

## 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?  → 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.
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. 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.
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.
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
36°

b. m∠DAE
108°

c. $$m \widehat{B D}$$

d. $$m \widehat{C E D}$$
180°

Exercise 2.
In circle B, AB = CD. Find the following angles of measure. a. $$m \widehat{C D}$$
60°

b. $$m \widehat{C A D}$$
300°

c. $$m \widehat{A C D}$$
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
20°

b. $$m \widehat{E C}$$
160°

c. $$m \widehat{D E C}$$
260°

Exercise 4.
Given circle A with m∠CAD = 37°, find the following angles of measure. a. $$m \widehat{C B D}$$
323°

b. m∠CBD
18.5°

c. m∠CED
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.

b. Name an inscribed angle.
∠CBD

c. Name a chord.
Answers will vary. $$\overline{B D}$$

d. Name a minor arc.
Answers will vary. $$\widehat{C D}$$

e. Name a major arc.
$$\widehat{C B D}$$

f. Find $$m \widehat{C D}$$
50°

g. Find $$m \widehat{C B D}$$.
310°

h. Find m∠CBD.
25°

Question 2.
Given circle A, find the measure of each minor arc. $$m \widehat{B E}$$ = 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. 100°

b. m∠CAB
80°

c. $$m \widehat{B C}$$
80°

d. $$m \widehat{B D}$$
100°

e. $$m \widehat{B C D}$$
260°

Question 4.
Find the measure of angle x. 33°

Question 5.
In the figure, m∠BAC = 126° and m∠BED = 32°. Find m∠DEC. 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 = _________________________________
$$m \widehat{B K}$$ = $$m \widehat{K C}$$ 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.

b. Name an inscribed angle.

c. Name a chord that is not a diameter.
Answers will vary. $$\overline{C E}$$

d. What is the measure of ∠CAD?
56°

e. What is the measure of ∠CBD?
g. What is the degree measure of $$\widehat{C D B}$$?