## Engage NY Eureka Math Geometry Module 1 Lesson 29 Answer Key

### Eureka Math Geometry Module 1 Lesson 29 Opening Exercise Answer Key

Opening Exercise:

Construct the midsegment of the triangle below. A midsegment is a line segment that joins the midpoints of two sides of a triangle or trapezoid. For the moment, we will work with a triangle.

a. Use your compass and straightedge to determine the midpoints of \(\overline{A B}\) and \(\overline{A C}\) as X and Y, respectively.

Answer:

b. Draw midsegment \(\overline{X Y}\).

Answer:

Compare ∠AXY and ∠ABC; compare ∠AYX and ∠ACB. Without using a protractor, what would you guess is the relationship between these two pairs of angles? What are the implications of this relationship?

Answer:

m∠AXY = m∠ABC, m∠AYX = m∠ACB; \(\overline{X Y}\) || \(\overline{B C}\)

Discussion:

Note that though we chose to determine the midsegment of \(\overline{A B}\) and \(\overline{A C}\), we could have chosen any two sides to work with. Let us now focus on the properties associated with a midsegment. The midsegment of a triangle is parallel to the third side of the triangle and half the length of the third side of the triangle.

We can prove these properties to be true. Continue to work with the figure from the Opening Exercise.

Given: \(\overline{X Y}\) is a midsegment of ΔABC.

Prove: \(\overline{X Y}\) || \(\overline{B C}\) and XY = \(\frac{1}{2}\)BC

Construct the following: In the Opening Exercise figure, draw ΔYGC according to the following steps. Extend \(\overline{X Y}\) to point G so that YG = XY. Draw \(\overline{G C}\).

Question (1)

What is the relationship between XV and YG? Explain why.

Answer:

Equal, by construction.

Question (2)

What is the relationship between m∠AYX and m∠GYC? Explain why.

Answer:

Equal, vertical angles are equal in measure.

Question (3)

What is the relationship between AY and YC? Explain why.

Answer:

Equal, Y is the midpoint of \(\overline{A C}\).

Question (4)

What is the relationship between A AXY and A CGY? Explain why.

Answer:

Congruent, SAS.

Question (5)

What is the relationship between GC and AX? Explain why.

Answer:

Equal, corresponding sides of congruent triangles are equal in length.

Question (6)

Since AX = BX, what other conclusion can be drawn? Explain why.

Answer:

GC = BX, substitution.

Question (7)

What is the relationship between m∠AXY and m∠CGV? Explain why.

Answer:

Equal, corresponding angles of congruent triangles are equal in measure.

Question (8)

Based on (7), what other conclusion can be drawn about \(\overline{A B}\) and \(\overline{G C}\)? Explain why.

Answer:

\(\overline{A B}\) || \(\overline{G C}\)

Question (9)

What conclusion can be drawn about BXGC based on (7) and (8)? Explain why.

Answer:

BXGC is a parallelogram; one pair of opposite sides is equal and parallel. Also, \(\overline{X Y}\) || \(\overline{B C}\).

Question (10)

Based on (9), what is the relationship between XG and BC?

Answer:

XG = BC, opposite sides of a parallelogram are equal.

Question (11)

Since YG = XV, XG = __ XV. Explain why.

Answer:

2, Substitution

Question (12)

This means BC = ___ XY. Explain why.

Answer:

2, Substitution

Question (13)

Or by division, XY = ___ B.

Answer:

\(\frac{1}{2}\).

Note that Steps (9) and (13) demonstrate our Prove statement.

### Eureka Math Geometry Module 1 Lesson 29 Exercise Answer Key

Apply what you know about the properties of midsegments to solve the following exercises.

Exercise 1.

x = ___

Perimeter of ∆ABC = _____

Answer:

x = 15

Perimeter of ∆ABC = 6

Exercise 2.

x = ___

y = ___

Answer:

x = 50°

y = 70°

Exercise 3.

In ∆RST, the midpoints of each side have been marked by points X, Y, and Z.

→ Mark the halves of each side divided by the midpoint with a congruency mark. Remember to distinguish congruency marks for each side.

→ Draw midsegments \(\overleftrightarrow{X Y}\), \(\overleftrightarrow{Y Z}\), and \(\overleftrightarrow{X Z}\). Mark each midsegment with the appropriate congruency mark from the sides of the triangle.

a. What conclusion can you draw about the four triangles within ∆RST? Explain why.

Answer:

All four are congruent. SSS

b. State the appropriate correspondences among the four triangles within ∆RST.

Answer:

∆RXY, ∆YZT, ∆XSZ, ∆ZYX

c. State a correspondence between RST and any one of the four small triangles.

Answer:

∆RXY, ∆RST

Exercise 4.

Find x.

Answer:

x = 9°

### Eureka Math Geometry Module 1 Lesson 29 Problem Set Answer Key

Exercises 1 – 4:

Use your knowledge of triangle congruence criteria to write proofs for each of the following problems.

Question 1.

\(\overline{W X}\) is a midsegment of ∆ABC, and \(\overline{Y Z}\) is a midsegment of ∆CWX. BX = AW

a. What can you conclude about ∠A and ∠B? Explain why.

Answer:

∠A ≅ ∠B, BX = AW, so CX = CW; the triangle is isosceles.

b. What is the relationship of the lengths \(\overline{Y Z}\) and \(\overline{A B}\)?

Answer:

YZ = \(\frac{1}{4}\)AB or 4YZ = AB

Question 2.

\(\overline{A B}\) || \(\overline{C D}\) and \(\overline{A D}\) || \(\overline{B C}\). W, X, Y, and Z are the midpoints of \(\overline{A D}\), \(\overline{A B}\) , \(\overline{B C}\), and \(\overline{C D}\), respectively. AD = 18, WZ = 11, and BX = 5. mLWAC = 33°, m∠XBY = 73°, m∠RYX = 74°, m∠DCA = 74°.

a. m∠DZW= ____

Answer:

74°

b. Perimeter of ABYW = ____

Answer:

38

c. Perimeter of ABCD = ____

Answer:

56

d. m∠WAX=____

Answer:

107°

m∠B= ____

Answer:

73°

m∠YCZ = ___

Answer:

107°

m∠D=____

Answer:

73°

e. What kind of quadrilateral is ABCD?

Answer:

Parallelogram

### Eureka Math Geometry Module 1 Lesson 29 Exit Ticket Answer Key

Use the properties of midsegments to solve for the unknown value in each question.

Question 1.

R and S are the midpoints of XW and WV, respectively.

What is the perimeter of ∆WXY?

Answer:

82

Question 2.

What is the perimeter of ∆EFG?

Answer:

62