Proofs about Parallelograms

ID: burik-sovop
Created by Illustrative MathIllustrative Math
Subject: Geometry
Grade: 9-12

Proofs about Parallelograms

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Problem 1

Conjecture: A quadrilateral with one pair of sides both congruent and parallel is a parallelogram.

1) Draw a diagram of the situation and mark the given information on the diagram.

2) Restate the conjecture as a specific statement using the diagram.

Problem 2

3) In quadrilateral ABCDABCD, ADAD is congruent to BCBC, and ADAD is parallel to BCBC. Show that ABCDABCD is a parallelogram.

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Problem 3

4) ABDEABDE is an isosceles trapezoid. Select all of the statements that could be used to show that the diagonals of an isosceles trapezoid are congruent. Write each corresponding letter in the answer box and separate letters with commas.

a) ABEBAD\triangle ABE \cong \triangle BAD \quad\quad b) ABEBDE\triangle ABE \cong \triangle BDE \quad\quad c) AEDBAD\triangle AED \cong \triangle BAD \quad\quad d) AEDBDE\triangle AED \cong \triangle BDE \quad\quad e) There are none.

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Problem 4

5) Select the conjecture with the rephrased statement of proof to show the diagonals of a parallelogram bisect each other.

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a) In parallelogram EFGH, show triangle HEF is congruent to triangle FGH.\text{In parallelogram } EFGH \text{, show triangle } HEF \text{ is congruent to triangle } FGH \text{.}b) In parallelogram EFGH, show triangle EKH is congruent to triangle GKF.\text{In parallelogram } EFGH \text{, show triangle } EKH \text{ is congruent to triangle } GKF \text{.}c) In parallelogram EFGH, show EK is congruent to KG and FK is congruent to KH.\text{In parallelogram } EFGH \text{, show } EK \text{ is congruent to } KG \text{ and } FK \text{ is congruent to } KH \text{.}d) In parallelogram EFGH with GH congruent to FE and EH congruent to FG, show EFGH is a parallelogram.\text{In parallelogram } EFGH \text{ with } GH \text{ congruent to } FE \text{ and } EH \text{ congruent to } FG \text{, show } EFGH \text{ is a parallelogram.}
Problem 5

6) Is triangle EJHEJH congruent to triangle EIHEIH?

Given: HJJE\overline{HJ} \perp \overline{JE}, HIIE\overline{HI} \perp \overline{IE}, HJHI\overline{HJ} \cong \overline{HI}

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True or false? Write below.

7) Explain your reasoning.

Problem 6

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

a) Segment DCDC is congruent to segment ABAB. \quad\quad b) Segment DADA is congruent to segment CBCB. \quad\quad c) Line DCDC is parallel to line ABAB. \quad\quad d) Line DADA is parallel to line CBCB. \quad\quad e) Angle CBECBE is congruent to angle DEADEA. \quad\quad f) Angle CEBCEB is congruent to angle DEADEA.

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Problem 7

9) Which conjecture is possible to prove?

a) All quadrilaterals with 4 equal angles are congruent.\text{All quadrilaterals with 4 equal angles are congruent.}b) All quadrilaterals with 4 equal sides are congruent.\text{All quadrilaterals with 4 equal sides are congruent.}c) All triangles with 3 equal angles are congruent.\text{All triangles with 3 equal angles are congruent.}d) All triangles with 3 equal sides are congruent.\text{All triangles with 3 equal sides are congruent.}