Proofs about Quadrilaterals

ID: tulot-tolur
Created by Illustrative MathIllustrative Math
Subject: Geometry
Grade: 9-12

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 EFGHEFGH is a parallelogram and angle HEFHEF is right, which reasoning about angles will help her prove that angle FGHFGH is also a right angle?

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

2) ABDEABDE 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 ABEABE and triangle DBEDBE \quad\quad b) Triangle ABDABD and triangle DAEDAE \quad\quad c) Triangle ABEABE and triangle BADBAD d) Triangle AEDAED and triangle BDEBDE \quad\quad e) Triangle EABEAB and triangle EDBEDB

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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.

IDStatement of Proof1In quadrilateral EFGH with GHcongruent to FE and EH congruent to FG, show EFGH is a parallelogram.2In parallelogram EFGH, show GH is congruent to FE and EH congruent to FG.3In quadrilateral EFGH with EKcongruent to KG and FK congruent to KH, show EFGH is a parallelogram.4In parallelogram EFGH, show EK is congruent to KG and FKcongruent to KH.\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}

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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 \quad\quad b) Corresponding congruent Side-Angle-Side \quad\quad c) Corresponding congruent Side-Side-Angle \quad\quad d) 3 congruent sides \quad\quad e) 2 congruent sides \quad\quad 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 EBEB is congruent to segment ADAD. \quad\quad b) Segment DCDC is congruent to segment ABAB. \quad\quad c) Segment DADA is congruent to segment CBCB. \quad\quad d) Angle CBECBE is congruent is angle ABEABE. \quad\quad e) Angle CEBCEB is congruent to angle DEADEA. \quad\quad f) Line DADA is parallel to line CBCB. \quad\quad g) Line DCDC is parallel to line ABAB.

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Problem 6

Diego states that diagonal WYWY bisects angles ZWXZWX and ZYXZYX.

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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°80\degree and 20°20\degree and side length 5.

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