Question

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s is the midpoint of \\( \overline { q r } \\) and \\( \overline { p s } \perp \overline { q r } \\). complete the proof that \\( \overline { p r } \cong \overline { p q } \\).

Explanation:

Step1: Given

Statement 1 and 2 are given in the problem.

Step2: Definition of perpendicular lines

Since \( \overline{PS}\perp\overline{QR}\), by the definition of perpendicular lines, \( \angle PSQ\) and \( \angle PSR\) are right angles. And all right angles are congruent, so \( \angle PSQ\cong\angle PSR\) (statement 3).

Step3: Definition of mid - point

Since \(S\) is the mid - point of \( \overline{QR}\), by the definition of a mid - point, \( \overline{QS}\cong\overline{RS}\) (statement 4).

Step4: Reflexive property of congruence

For any segment \( \overline{AB}\), \( \overline{AB}\cong\overline{AB}\). So, \( \overline{PS}\cong\overline{PS}\) (statement 5) by the reflexive property of congruence.

Step5: SAS (Side - Angle - Side) congruence criterion

In \( \triangle PQS\) and \( \triangle PRS\), we have \( \overline{QS}\cong\overline{RS}\) (side), \( \angle PSQ\cong\angle PSR\) (angle), and \( \overline{PS}\cong\overline{PS}\) (side). So, \( \triangle PQS\cong\triangle PRS\) (statement 6) by the SAS (Side - Angle - Side) congruence criterion.

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

Since \( \triangle PQS\cong\triangle PRS\), then their corresponding sides \( \overline{PR}\) and \( \overline{PQ}\) are congruent. So, \( \overline{PR}\cong\overline{PQ}\) (statement 7) by CPCTC.

Answer:

1. Given
2. Given
3. Definition of perpendicular lines (all right angles are congruent)
4. Definition of mid - point
5. Reflexive property of congruence
6. SAS (Side - Angle - Side)
7. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)