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# Proving Congruence in Parallelogram RSTU: Applying Geometric Properties

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Question: Parallelogram RSTU is shown below. Ilan needs to prove that RS¯¯¯¯¯≅TU¯¯¯¯¯ and RU¯¯¯¯¯≅TS¯¯¯¯¯ . His proof is shown in the table below. Step Statement 1 RSTU is a parallelogram 2 RS¯¯¯¯¯ is parallel to TU¯¯¯¯¯ and RU¯¯¯¯¯ is parallel to TS¯¯¯¯¯ 3 ∠RSU≅∠TUS and ∠RUS≅∠TSU 4 SU¯¯¯¯¯≅US¯¯¯¯¯ 5 △RSU≅△TUS 6 RS¯¯¯¯¯≅TU¯¯¯¯¯ and RU¯¯¯¯¯≅TS¯¯¯¯¯ Select ALL reasons that support one or more statements in the proof. symmetric property definition of a parallelogram alternate interior angles are congruent opposite sides of a parallelogram are congruent corresponding parts of congruent triangles are congruent

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The reasons that support one or more statements in the proof are: - Definition of a parallelogram (supports statement 2: RS is parallel to TU and RU is parallel to TS) - Alternate interior angles are congruent (supports statement 3: ∠RSU ≅ ∠TUS and ∠RUS ≅ ∠TSU due to the parallel sides of the parallelogram and transversal lines) - Corresponding parts of congruent triangles are congruent (supports statement 6: After proving △RSU ≅ △TUS in statement 5, we can then say that RS ≅ TU and RU ≅ TS based on this property) - Opposite sides of a parallelogram are congruent (supports statement 1 and indirectly leads to statement 6: Since RSTU is a parallelogram, RS and UT are congruent as are UR and TS). So four of the five options you've provided are reasons relevant to the proof. The only irrelevant one is symmetric property, which is a property of equality that doesn't apply to any step in the proof.

Jan. 12, 2024, 8:22 a.m.

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