Question

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s is the midpoint of \\( \overline{qr} \\) and \\( \overline{pr} \cong \overline{pq} \\). complete the proof that \\( \angle q \cong \angle r \\).

Explanation:

Step1: Given information

Statement 1 is given in the problem. Statement 2 is also given in the problem.

Step2: Mid - point property

Since \(S\) is the mid - point of \(\overline{QR}\), by the definition of a mid - point, \(QS = RS\), so \(\overline{QS}\cong\overline{RS}\) (definition of congruent segments).

Step3: Reflexive property

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

Step4: SSS (Side - Side - Side) congruence criterion

We have \(\overline{PR}\cong\overline{PQ}\) (given), \(\overline{QS}\cong\overline{RS}\) (from mid - point), and \(\overline{PS}\cong\overline{PS}\) (reflexive property). So, \(\triangle PQS\cong\triangle PRS\) by the SSS (Side - Side - Side) congruence criterion.

Step5: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Since \(\triangle PQS\cong\triangle PRS\), then \(\angle Q\cong\angle R\) by CPCTC (Corresponding Parts of Congruent Triangles are Congruent).

Answer:

1. Given
2. Given
3. Definition of mid - point
4. Reflexive property of congruence
5. SSS (Side - Side - Side)
6. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)