Question

∠qpr ≅ ∠pqs, ( overline{qs} cong overline{pr} ), and ( overline{pq} parallel overline{rs} ). complete the proof that ( \triangle qrs cong \triangle psr ).

(image of a quadrilateral with points p, q, r, s and intersection t)

	statement	reason
1	( angle qpr cong angle pqs )	given
2	( overline{qs} cong overline{pr} )	given
3	( overline{pq} parallel overline{rs} )	given
4	( angle qpr cong angle prs )
5	( angle qsr cong angle pqs )	alternate interior angles theorem
6	( angle pqs cong angle prs )
7	( angle qsr cong angle prs )
8	( overline{rs} cong overline{rs} )	reflexive property of congruence
9	( \triangle qrs cong \triangle psr )	sas

Explanation:

Step1: Analyze Step 4

Since \(\overline{PQ} \parallel \overline{RS}\) (given in step 3), and we are looking at \(\angle QPR\) and \(\angle PRS\), these are alternate interior angles formed by the transversal \(\overline{PR}\) intersecting the parallel lines \(\overline{PQ}\) and \(\overline{RS}\). So the reason for \(\angle QPR \cong \angle PRS\) is the Alternate Interior Angles Theorem.

Step2: Analyze Step 6

We know from step 1 that \(\angle QPR \cong \angle PQS\) and from step 4 that \(\angle QPR \cong \angle PRS\). By the Transitive Property of Congruence (if \(a \cong b\) and \(a \cong c\), then \(b \cong c\)), we can conclude \(\angle PQS \cong \angle PRS\).

Step3: Analyze Step 7

From step 5, \(\angle QSR \cong \angle PQS\), and from step 6, \(\angle PQS \cong \angle PRS\). Using the Transitive Property of Congruence again, we get \(\angle QSR \cong \angle PRS\).

Answer:

• Step 4 Reason: Alternate Interior Angles Theorem
• Step 6 Reason: Transitive Property of Congruence
• Step 7 Reason: Transitive Property of Congruence