High School

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PROVING A THEOREM Use the diagram of PQRS

with the auxiliary line segment drawn to prove the

Opposite Sides Parallel and Congruent Theorem

(Theorem 7.9).

Given QR|PS, QR = PS

Prove PQRS is a parallelogram.

PROVING A THEOREM Use the diagram of PQRS with the auxiliary line segment drawn to prove the Opposite Sides Parallel and Congruent Theorem Theorem 7

Answer :

To prove quadrilateral PQRS is a parallelogram: Given QR parallel to PS, QR congruent to PS. Construct auxiliary line PR. Show corresponding angles and sides congruent. PQRS is a parallelogram.

To prove that the quadrilateral PQRS is a parallelogram, we can use the Opposite Sides Parallel and Congruent Theorem, which states that if a quadrilateral has one pair of opposite sides that are both parallel and congruent, then it is a parallelogram.

Given:

1. QR is parallel to PS.

2. QR is congruent to PS.

To prove:

PQRS is a parallelogram.

Proof:

1. QR is parallel to PS (Given).

2. QR is congruent to PS (Given).

3. Construct auxiliary line segment PR.

4. ∠QPR ≅ ∠SPR (Corresponding angles formed by parallel lines QR and PS).

5. ∠PRQ ≅ ∠PRS (Corresponding angles formed by parallel lines QR and PS).

6. PQ is congruent to PQ (Reflexive property).

7. ΔPQR is congruent to ΔPSR (ASA congruence, using steps 2, 4, and 5).

8. QR is congruent to PS (Corresponding parts of congruent triangles).

9. PQ is parallel to SR (If corresponding sides of two triangles are congruent, then the triangles are congruent, and the corresponding angles are equal).

10. Thus, PQRS is a parallelogram (Opposite sides parallel and congruent theorem).

Therefore, PQRS is a parallelogram.

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