# Eureka Math Geometry Module 5 Lesson 6 Answer Key

## Engage NY Eureka Math Geometry Module 5 Lesson 6 Answer Key

### Eureka Math Geometry Module 5 Lesson 6 Exercise Answer Key

Opening Exercise
In a circle, a chord $$\overline{D E}$$ and a diameter $$\overline{A B}$$ are extended outside of the circle to meet at point C. If mâˆ DAE = 46Â°, and mâˆ DCA = 32Â°, find mâˆ DEA.

Let mâˆ DEA = yËš, mâˆ EAB = xËš

In â–³ABD, mâˆ DBA =
Reason:
In â–³ABD, mâˆ DBA = yËš
Reason: angles inscribed in same arc are congruent

Reason:
mâˆ ADB = 90Â° Reason: angle inscribed in semicircle

âˆ´46 + x + y + 90 =
Reason:
âˆ´46 + x + y + 90 = 180 Reason: sum of angles of triangle is 180Â°

x + y =
x + y = 44

In â–³ACE, y = x + 32
Reason:
In â–³ACE, y = x + 32
Reason: Exterior angle of a triangle is equal to the sum of the remote interior angles

x + x + 32 =
Reason:
x + x + 32 = 44
Reason: substitution

x =
x = 6

y =
y = 38

mâˆ DEA =
mâˆ DEA = 38Â°

Exercises
Find the value of x in each figure below, and describe how you arrived at the answer.

Exercise 1.
Hint: Thalesâ€™ theorem

mâˆ BEC = 90Â° inscribed in a semicircle
mâˆ EBC = mâˆ ECB = 45Â° base angles of an isosceles triangle are congruent and sum of angles of a triangle = 180Â°
mâˆ EBC = mâˆ EDC = 45Â° angles inscribed in the same arc are congruent
x = 45

Exercise 2.

mâˆ BAD = 146Ëš, if parallel lines cut by a transversal, then interior angles on the same side are supplementary. Then the $$m \widehat{B D}$$ = 146Ëš, because âˆ BAD is a central angle intercepting $$\widehat{B D}$$. Then remaining arc of the circle, $$\widehat{B C D}$$, has a measure of 214Â°. Then mâˆ BED = 107Ëš since it is an inscribed angle intercepting $$\widehat{B C D}$$. The angle sum of a quadrilateral is 360Â°, which means x = 73.

Exercise 3.

mâˆ BEC = mâˆ CFB = $$\frac{1}{2}$$ mâˆ BAC = 52Â°
Inscribed angles are half the measure of the central angle intercepting the same arc.
mâˆ DEG = 128Â° linear pair with âˆ BEC
mâˆ GFD = 128Â° linear pair with âˆ CFB
mâˆ EGF = 74Â° sum of angles of a quadrilateral
x = 74 vertical angles

Exercise 4.

The measures of arcs $$\widehat{D E}$$, $$\widehat{E F}$$, and $$\widehat{F C}$$ are each 60Ëš, since the intercepted arc of an inscribed angle is double the measure of the angle. This means $$\widehat{m D E} C$$ = 180Ëš, or $$\widehat{D E C}$$ is a semicircle. This means x is 90, since âˆ DBC is inscribed in a semicircle.

### Eureka Math Geometry Module 5 Lesson 6 Problem Set Answer Key

In Problems 1â€“5, find the value x.
Question 1.

x = 40.5

Question 2.

x = 57

Question 3.

x = 15

Question 4.

x = 34

Question 5.

x = 90

Question 6.
If BF = FC, express y in terms of x.

y = 90 – $$\frac{x}{2}$$

Question 7.
a. Find the value of x.

x = 90

b. Suppose the mâˆ C = aÂ°. Prove that mâˆ DEB = 3aÂ°.
mâˆ D = aÂ° (alternate angles are equal in measure), mâˆ A = 2aÂ° (inscribed angles half the central angle), aÂ° + 2aÂ° + mâˆ AED = 180Â° (the sum of the angles of triangle is 180Â°), mâˆ AED = (180 – 3a)Â°, mâˆ AED + mâˆ DEB = 180Â° (angles form line), (180 – 3a)Â° + mâˆ DEB = 180Â° (substitution), mâˆ DEB = 3aÂ°

Question 8.
In the figure below, three identical circles meet at B, F, C, and E, respectively. BF = CE. A, B, C and F, E, D lie on straight lines.
Prove ACDF is a parallelogram.

PROOF:

Join BE and CF.
BF = CE Reason: ______________________________
a = __________ = __________ = __________ = d Reason: ______________________________
__________ = __________ Alternate interior angles are equal in measure.
$$\overline{A C}$$ âˆ¥ $$\overline{F D}$$
__________ = __________ Corresponding angles are equal in measure.
$$\overline{A F}$$ âˆ¥ $$\overline{B E}$$
__________ = __________ Corresponding angles are equal in measure.
$$\overline{B E}$$ âˆ¥ $$\overline{C D}$$
$$\overline{A F}$$ âˆ¥ $$\overline{B E}$$âˆ¥$$\overline{C D}$$
ACDF is a parallelogram.
Join BE and CF.
BF = CEÂ  Â  Â  Â Reason: Given
a = b = f = e = dÂ  Â  Â  Â Reason: Angles inscribed in congruent arcs are equal in
mâˆ CBE = mâˆ FEBÂ  Â  Â  Â Alternate interior angles are equal in measure.
$$\overline{A C}$$ âˆ¥ $$\overline{F D}$$
mâˆ A = mâˆ CBEÂ  Â  Â  Â  Â Corresponding angles are equal in measure.
$$\overline{A F}$$ âˆ¥ $$\overline{B E}$$
mâˆ D = mâˆ BEFÂ  Â  Â  Â  Â  Â Corresponding angles are equal in measure.
$$\overline{B E}$$ âˆ¥ $$\overline{C D}$$
$$\overline{A F}$$ âˆ¥ $$\overline{B E}$$âˆ¥$$\overline{C D}$$
ACDF is a parallelogram.

### Eureka Math Geometry Module 5 Lesson 6 Exit Ticket Answer Key

Question 1.
Find the measure of angles x and y. Explain the relationships and theorems used.

mâˆ EAC = 42Â° (linear pair with âˆ BAE). mâˆ EFC = $$\frac{1}{2}$$ mâˆ EAC = 21Â° (inscribed angle is half measure of central angle with same intercepted arc). x = 21.