# Eureka Math Geometry Module 1 Lesson 27 Answer Key

## Engage NY Eureka Math Geometry Module 1 Lesson 27 Answer Key

### Eureka Math Geometry Module 1 Lesson 27 Exercise Answer Key

Exercises

Exercise 1.
Given: AB=AC, RB=RC
Prove: SB=SC

AB=AC, RB=RC Given
AR=AR Reflexive property
â–³ARCâ‰… â–³ARB SSS
mâˆ ARC=mâˆ ARB Corresponding angles of congruent triangles are equal in measure.
mâˆ ARC+mâˆ SRC=180Â°,
mâˆ ARB+mâˆ SRB=180Â° Linear pairs form supplementary angles.
mâˆ SRC=mâˆ SRB Angles supplementary to either the same angle or to congruent angles are equal in measure.
SR=RS Reflexive property
â–³SRBâ‰… â–³SRC SAS
SB=SC Corresponding sides of congruent angles are equal in length.

Exercise 2.
Given: Square ABCS â‰… Square EFGS,
$$\overleftrightarrow{\boldsymbol{R A B}}$$, $$\overleftrightarrow{\boldsymbol{R E F}}$$
Prove: â–³ASRâ‰… â–³ESR

Square ABCS â‰… Square EFGS Given
AS=ES Corresponding sides of congruent squares are equal in length.
SR=RS Reflexive property
âˆ BAS and âˆ FES are right angles. Definition of square
âˆ BAS and âˆ SAR form a linear pair. Definition of linear pair
âˆ FES and âˆ SER form a linear pair. Definition of linear pair
âˆ SAR and âˆ SER are right angles. Two angles that are supplementary and congruent each measure 90Â° and are, therefore, right angles.
â–³ASR and â–³ESR are right triangles. Definition of right triangle
â–³ASRâ‰… â–³ESR HL

Exercise 3.
Given: JK=JL, JX=JY
Prove: KX=LY

JX=JY Given
mâˆ JXY=mâˆ JYX Base angles of an isosceles triangle are equal in measure.
mâˆ JXK+mâˆ JXY=180Â°,
mâˆ JYL+mâˆ JYX=180Â° Linear pairs form supplementary angles.
mâˆ JXK+mâˆ JXY=mâˆ JYL+mâˆ JYX Substitution property of equality
mâˆ JXK+mâˆ JXY=mâˆ JYL+mâˆ JXY Substitution property of equality
mâˆ JXK=mâˆ JYL Angles supplementary to either the same angle or congruent angles are equal in measure.
JK=JL Given
mâˆ K=mâˆ L Base angles of an isosceles triangle are equal in measure.
â–³JXKâ‰… â–³JYL AAS
KX=LY Corresponding sides of congruent triangles are equal in length.

Exercise 4.
Given: $$\overline{A D}$$ âŠ¥$$\overline{D R}$$, $$\overline{A B}$$ âŠ¥$$\overline{B R}$$,
$$\overline{A D}$$ â‰… $$\overline{A B}$$
Prove: âˆ DCRâ‰…âˆ BCR

$$\overline{A D}$$ âŠ¥$$\overline{D R}$$, $$\overline{A B}$$ âŠ¥$$\overline{B R}$$ Given
â–³ADR and â–³ABR are right triangles. Definition of right triangle
$$\overline{A D}$$ â‰… $$\overline{A B}$$ Given
$$\overline{A R}$$ â‰… $$\overline{A R}$$ Reflexive property
âˆ ARDâ‰…ARB Corresponding angles of congruent triangles are congruent.
mâˆ ARD+mâˆ DRC=180Â°,
mâˆ ARB+mâˆ BRC=180Â° Linear pairs form supplementary angles.
mâˆ ARD+mâˆ DRC=mâˆ ARB+mâˆ BRC Transitive property
mâˆ DRC=mâˆ BRC Angles supplementary to either the same angle or congruent angles are equal in measure.
$$\overline{D R}$$ â‰… $$\overline{B R}$$ Corresponding sides of congruent triangles are congruent.
$$\overline{R C}$$ â‰…$$\overline{R C}$$ Reflexive property
â–³DRCâ‰… â–³BRC SAS
âˆ DRCâ‰…âˆ BRC Corresponding angles of congruent triangles are congruent.

Exercise 5.
Given: AR=AS, BR=CS,
$$\overline{R X}$$ âŠ¥$$\overline{A B}$$, $$\overline{S Y}$$ âŠ¥$$\overline{A C}$$
Prove: BX=CY

AR=AS Given
mâˆ ARS=mâˆ ASR Base angles of an isosceles triangle are equal in measure.
mâˆ ARS+mâˆ ARB=180Â°,
mâˆ ASR+mâˆ ASC=180Â° Linear pairs form supplementary angles.
mâˆ ARS+mâˆ ARB=mâˆ ASR+mâˆ ASC Transitive property
mâˆ ARB=mâˆ ASC Subtraction
BR=CS Given
â–³ARBâ‰… â–³ASC SAS
âˆ ABRâ‰…âˆ ACS Corresponding angles of congruent triangles are congruent.
$$\overline{R X}$$ âŠ¥$$\overline{A B}$$, $$\overline{S Y}$$ âŠ¥$$\overline{A C}$$ Given
mâˆ RXB=90Â°=mâˆ SYC Definition of perpendicular line segments
â–³BRXâ‰… â–³SYC AAS
BX=CY Corresponding sides of congruent triangles are equal in length.

Exercise 6.
Given: AX=BX, mâˆ AMB=mâˆ AYZ=90Â°
Prove: NY=NM

AX=BX Given
mâˆ AMB=mâˆ AYZ=90Â° Given
mâˆ BXM=mâˆ AXY Vertical angles are equal in measure.
â–³BXMâ‰…â–³AXY AAS
XM=XY Corresponding sides of congruent triangles are equal in length.
BY=AM Substitution property of equality
mâˆ BYN=90Â° Vertical angles are equal in measure.
mâˆ AMB+mâˆ AMN=180Â° Linear pairs form supplementary angles.
mâˆ AMN=90Â° Subtraction property of equality
mâˆ MNY=mâˆ MNY Reflexive property
â–³BYNâ‰… â–³AMN AAS
NY=NM Corresponding sides of congruent triangles are equal in length.

### Eureka Math Geometry Module 1 Lesson 27 Problem Set Answer Key

Use your knowledge of triangle congruence criteria to write a proof for the following:
In the figure $$\overline{B E}$$ â‰… $$\overline{C E}$$, $$\overline{D C}$$ âŠ¥$$\overline{A B}$$, and $$\overline{B E}$$ âŠ¥$$\overline{A C}$$ ; prove $$\overline{A E}$$ â‰… $$\overline{R E}$$.

mâˆ ERC=mâˆ BRD Vertical angles are equal in measure.
$$\overline{D C}$$ âŠ¥$$\overline{A B}$$, $$\overline{B E}$$ âŠ¥$$\overline{A C}$$ Given
mâˆ BDR=90Â°, mâˆ REC=90Â° Definition of perpendicular lines
mâˆ ABE=mâˆ RCE Sum of the angle measures in a triangle is 180Â°.
mâˆ BAE=mâˆ BRD Sum of the angle measures in a triangle is 180Â°.
mâˆ BAE=mâˆ ERC Substitution property of equality
$$\overline{B E}$$ â‰… $$\overline{C E}$$ Given
â–³BAEâ‰… â–³CRE AAS
$$\overline{A E}$$ â‰…$$\overline{R E}$$ Corresponding sides of congruent triangles are congruent.

### Eureka Math Geometry Module 1 Lesson 27 Exit Ticket Answer Key

Given: M is the midpoint of $$\overline{G R}$$, âˆ G â‰…âˆ R
Prove: â–³GHMâ‰… â–³RPM

M is the midpoint of $$\overline{G R}$$. Given