# Proofs about Quadrilaterals

ID: tulot-tolur
Illustrative Mathematics
Subject: Geometry
Grade: 9-12
PreviewAnswer Key

12 questions

# Proofs about Quadrilaterals

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##### Problem 1

1) Lin is using the diagram to prove the statement, “If a parallelogram has one right angle, it is a rectangle.” Given that ﻿$EFGH$﻿ is a parallelogram and angle ﻿$HEF$﻿ is right, which reasoning about angles will help her prove that angle ﻿$FGH$﻿ is also a right angle?

a) $\text{Corresponding angles are congruent when parallel lines are cut by a transversal.}$b) $\text{Opposite angles in a parallelogram are congruent.}$c) $\text{Vertical angles are congruent.}$d) $\text{The base angles of an isosceles triangle are congruent.}$
##### Problem 2

2) ﻿$ABDE$﻿ is an isosceles trapezoid. Select all pairs of congruent triangles. Write each corresponding letter in the answer box and separate letters with commas.

a) Triangle ﻿$ABE$﻿ and triangle ﻿$DBE$﻿ b) Triangle ﻿$ABD$﻿ and triangle ﻿$DAE$﻿ c) Triangle ﻿$ABE$﻿ and triangle ﻿$BAD$﻿

d) Triangle ﻿$AED$﻿ and triangle ﻿$BDE$﻿ e) Triangle ﻿$EAB$﻿ and triangle ﻿$EDB$﻿

##### Problem 3

Match each conjecture with the rephrased statement of proof connected to the diagram. Write the number of the corresponding statement in the answer box.

﻿$\begin{array}{|c|l|} \hline \\[-1em] \textbf{ID} & \textbf{Statement of Proof} \\[-1em] \\ \hline \\[-1em] 1 & \text{In quadrilateral } EFGH \text{ with } GH \\ & \text{congruent to } FE \text{ and } EH \text{ congruent to } \\ & FG, \text{ show } EFGH \text{ is a parallelogram.} \\[-1em] \\ \hline \\[-1em] 2 & \text{In parallelogram }EFGH, \text{ show } GH \text{ is } \\ & \text{congruent to } FE \text{ and } EH \text{ congruent to }\\ & FG. \\[-1em] \\ \hline \\[-1em] 3 & \text{In quadrilateral } EFGH \text{ with } EK \\ & \text{congruent to } KG \text{ and } FK \text{ congruent to } \\ & KH, \text{ show } EFGH \text{ is a parallelogram.} \\[-1em] \\ \hline \\[-1em] 4 & \text{In parallelogram }EFGH, \text{ show } EK \text{ is } \\ & \text{congruent to } KG \text{ and } FK \text{congruent to }\\ & KH. \\[-1em] \\ \hline \end{array}$﻿

3) The diagonals of a parallelogram bisect each other.

4) In a parallelogram, opposite sides are congruent.

5) A quadrilateral with opposite sides congruent is a parallelogram.

6) If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.

##### Problem 4

7) Which of the following criteria always proves triangles congruent? Select all that apply. Write each corresponding letter in the answer box and separate letters with commas.

a) Corresponding congruent Angle-Side-Angle b) Corresponding congruent Side-Angle-Side

c) Corresponding congruent Side-Side-Angle d) 3 congruent sides e) 2 congruent sides

f) 3 congruent angles

##### Problem 5

8) Select all true statements based on the diagram. Write each corresponding letter in the answer box and separate letters with commas.

a) Segment ﻿$EB$﻿ is congruent to segment ﻿$AD$﻿. b) Segment ﻿$DC$﻿ is congruent to segment ﻿$AB$﻿.

c) Segment ﻿$DA$﻿ is congruent to segment ﻿$CB$﻿. d) Angle ﻿$CBE$﻿ is congruent is angle ﻿$ABE$﻿.

e) Angle ﻿$CEB$﻿ is congruent to angle ﻿$DEA$﻿. f) Line ﻿$DA$﻿ is parallel to line ﻿$CB$﻿.

g) Line ﻿$DC$﻿ is parallel to line ﻿$AB$﻿.

##### Problem 6

Diego states that diagonal ﻿$WY$﻿ bisects angles ﻿$ZWX$﻿ and ﻿$ZYX$﻿.

9) Is he correct?

True or false? Write below.

10) Explain your reasoning.

##### Problem 7

11) Sketch the unique triangles that can be made with angle measures ﻿$80\degree$﻿ and ﻿$20\degree$﻿ and side length 5.

12) How do you know you have sketched all possibilities?