Question

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t is the midpoint of \\( \overline{rs} \\) and \\( \overline{qr} \cong \overline{qs} \\). complete the proof that \\( \angle sqt \cong \angle rqt \\).

Explanation:

Step1: Definition of midpoint

Since \(T\) is the midpoint of \(\overline{RS}\), by the definition of a midpoint, it divides the segment into two equal parts. So, \(\overline{RT}\cong\overline{ST}\).

Step2: Given

The problem statement gives \(\overline{QR}\cong\overline{QS}\).

Step3: Reflexive property

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

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

We have \(\overline{QR}\cong\overline{QS}\), \(\overline{RT}\cong\overline{ST}\), and \(\overline{QT}\cong\overline{QT}\). So, by the SSS (Side - Side - Side) congruence criterion, \(\triangle QRT\cong\triangle QST\).

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

Since \(\triangle QRT\cong\triangle QST\), their corresponding parts are congruent. So, \(\angle SQT\cong\angle RQT\) by CPCTC.

Answer:

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