## Engage NY Eureka Math Geometry Module 5 Lesson 8 Answer Key

### Eureka Math Geometry Module 5 Lesson 8 Exercise Answer Key

Opening Exercise
Given circle A with $$\overline{B C}$$ ⊥ $$\overline{D E}$$, FA = 6, and AC = 10. Find BF and DE. Explain your work. BF = 4, DE = 16.
$$\overline{A B}$$ is a radius with a measure of 10. If FA = 6, then BF = 10 – 6 = 4.
Connect $$\overline{A D}$$ and $$\overline{A E}$$. In △DAE, AD and AE are both equal to 10. Both △DFA and △EFA are right triangles and congruent, so by the Pythagorean theorem,
DF = FE = 8, making DE = 16.

Exercises

Exercise 1.
Given circle A with $$m \widehat{B C}$$ = 54° and ∠CDB ≅ ∠DBE, find $$m \widehat{D E}$$. Explain your work. $$m \widehat{D E}$$ = 54°. m∠CAB = 54° because the central angle has the same measure as its subtended arc. m∠CDB is 27° because an inscribed angle has half the measure of the central angle with the same inscribed arc. Since ∠CDB is congruent to ∠DBE, $$m \widehat{D E}$$ is 54° because it is double the angle inscribed in it.

Exercise 2.
If two arcs in a circle have the same measure, what can you say about the quadrilateral formed by the four endpoints? Explain.
If the arcs are congruent, their endpoints can be joined to form chords that are parallel ($$\overline{B C}$$ ∥$$\overline{D E}$$).
The chords subtending the congruent arcs are congruent ($$\overline{B D}$$ ≅ $$\overline{C E}$$).
A quadrilateral with one pair of opposite sides parallel and the other pair of sides congruent is an isosceles trapezoid.

Exercise 3.
Find the angle measure of $$\widehat{C D}$$ and $$\widehat{E D}$$. $$m \widehat{C D}$$ = 130°, $$m \widehat{E D}$$ = 50°

Exercise 4.
$$m \widehat{C B}$$ = $$m \widehat{E D}$$ and $$m \widehat{B C}$$ : $$m \widehat{B D}$$ : $$m \widehat{E C}$$ = 1 : 2 : 4. Find the following angle measures. a. m∠BCF
45°

b. m∠EDF
90°

c. m∠CFE
135°

Exercise 5.
$$\overline{B C}$$ is a diameter of circle A. $$m \widehat{B D}$$ : $$m \widehat{D E}$$ : $$m \widehat{E C}$$ = 1 : 3 : 5. Find the following arc measures. a. $$m \widehat{B D}$$
20°

b. $$m \widehat{D E C}$$
160°

c. $$m \widehat{E C B}$$
280°

### Eureka Math Geometry Module 5 Lesson 8 Problem Set Answer Key

Question 1.
Find the following arc measures. a. $$m \widehat{C E}$$
70°

b. $$m \widehat{B D}$$
70°

c. $$m \widehat{E D}$$
40°

Question 2.
In circle A, $$\overline{B C}$$ is a diameter, $$m \widehat{C E}$$ = $$m \widehat{E D}$$, and m∠CAE = 32°. 64°

58°

Question 3.
In circle A, ($$\overline{B C}$$ is a diameter, 2$$m \widehat{C E}$$ = $$m \widehat{E D}$$, and $$\overline{B C}$$ ∥ $$\overline{D E}$$. Find m∠CDE. 22.5°

Question 4.
In circle A, $$\overline{B C}$$ is a diameter and $$m \widehat{C E}$$ = 68°. a. Find $$m \widehat{C D}$$.
68°

b. Find m∠DBE.
68°

c. Find m∠DCE.
112°

Question 5.
In the circle given, $$\widehat{B C}$$ ≅ $$\widehat{E D}$$. Prove $$\overline{B E}$$ ≅ $$\overline{D C}$$. Join $$\overline{C E}$$.
BC = ED (congruent arcs have chords equal in length)
m∠CBE = m∠EDC (angles inscribed in same arc are equal in measure) m∠BCE = m∠DEC (angles inscribed in congruent arcs are equal in measure)
△BCE≅ △DEC (AAS)
$$\overline{B E}$$ ≅ $$\overline{D C}$$(corresponding sides of congruent triangles are congruent)

Question 6.
Given circle A with $$\overline{A D}$$ ∥ $$\overline{C E}$$, show ($$m \widehat{B D}$$ ≅ $$\widehat{D E}$$. Join $$\overline{B D}$$, $$\overline{D E}$$, $$\overline{A E}$$.
∠AEC ≅ ∠ACE, ∠AED ≅ ∠ADE, ∠ADB ≅ ∠ABD (base angles of isosceles triangles are congruent)
∠AEC ≅ ∠EAD (alternate interior angles are congruent)
m∠ADE + m∠DEA + m∠EAD = 180° (sum of angles of a triangle)
3m∠AED = 180° (substitution)
m∠AED = 60°; △BAD ≅ △DAE ≅ △EAC (SAS)
BD = DE (corresponding parts of congruent triangles)
$$m \widehat{B D}$$ ≅ $$m \widehat{D E}$$ (arcs subtended by congruent chords)

Question 7.
In circle A, $$\overline{A B}$$ is a radius, $$m \widehat{B C}$$ ≅ $$m \widehat{B D}$$, and m∠CAD = 54°. Find m∠ABC. Complete the proof. BC = BD _____________________

m∠________ = m∠_______          _____________________

2m∠________ + 54° = 360° ________

m∠BAC = ________

AB = AC ________

m∠________ = m∠ ________      __________________

2m∠ABC + m∠BAC = ________ Sum of ____________________

m∠ABC = ________
BC = BD Chords of congruent arcs

m∠BAC = m∠BAD      Angles inscribed in congruent arcs are equal in measure.

2m∠BAC + 54° = 360°           Circle

m∠BAC = 153°

m∠ABC = m∠ACB      Base angles of isosceles

2m∠ABC + m∠BAC = 180° Sum of angles of a triangle equal 180°

m∠ABC = 13.5°

### Eureka Math Geometry Module 5 Lesson 8 Exit Ticket Answer Key

Question 1.
Given circle A with radius 10, prove BE = DC. $$m \widehat{B E}$$ = $$m \widehat{D C}$$ (arcs are equal in degree measure to their inscribed central angles)
Given the circle at right, find $$m \widehat{B D}$$. 