Question

pq || rs. complete the proof that ∠q ≅ ∠s.
statement
1 pq || rs
2 pq ≅ rs
3 ∠prs ≅ ∠qpr
4 pr ≅ pr
5 △pqr ≅ △rsp
6 ∠q ≅ ∠s

Explanation:

Step1: Given

The problem states that \(\overline{PQ}\parallel\overline{RS}\), so this is the given information.

Step2: Given

From the figure, we can see that \(\overline{PQ}\cong\overline{RS}\) (marked as equal in the diagram), so this is given.

Step3: Alternate - interior angles

Since \(\overline{PQ}\parallel\overline{RS}\) and \(\overline{PR}\) is a transversal, by the alternate - interior angles theorem, \(\angle PRS\cong\angle QPR\).

Step4: Reflexive property

For any segment \(\overline{PR}\), \(\overline{PR}\cong\overline{PR}\) by the reflexive property of congruence (a segment is congruent to itself).

Step5: SAS (Side - Angle - Side)

We have \(\overline{PQ}\cong\overline{RS}\) (side), \(\angle QPR\cong\angle PRS\) (angle), and \(\overline{PR}\cong\overline{PR}\) (side). So, by the SAS (Side - Angle - Side) congruence criterion, \(\triangle PQR\cong\triangle RSP\).

Step6: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Since \(\triangle PQR\cong\triangle RSP\), then their corresponding parts \(\angle Q\) and \(\angle S\) are congruent.

Answer:

1. Given; 2. Given; 3. Alternate - interior angles theorem; 4. Reflexive property; 5. SAS (Side - Angle - Side); 6. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)