Question

1. given: r is the mid - point of qt, qs≅ts. prove: ∠rqs≅∠rts. statement: r is the mid - point of qt, qs≅ts, qr≅tr, rs≅rs, △rsq≅△rts, ∠rqs≅∠rts. reason: given, given, def of midpoint, reflexive property, sss, cpctc

Explanation:

Step1: Recall mid - point definition

Since R is the mid - point of QT, by the definition of a mid - point, we have QR = TR.

Step2: Use given side - equality

We are given that QS = TS and RS = RS (reflexive property).

Step3: Apply SSS congruence criterion

In \(\triangle RSQ\) and \(\triangle RTS\), we have QR = TR, QS = TS, and RS = RS. So, \(\triangle RSQ\cong\triangle RTS\) by the Side - Side - Side (SSS) congruence criterion.

Step4: Use CPCTC

Since \(\triangle RSQ\cong\triangle RTS\), corresponding parts of congruent triangles are congruent (CPCTC). So, \(\angle RQS=\angle RTS\).

Answer:

1. Statement: R is the mid - point of QT; Reason: Given
2. Statement: QS = TS; Reason: Given
3. Statement: RS = RS; Reason: Reflexive Property
4. Statement: QR = TR; Reason: Definition of mid - point
5. Statement: \(\triangle RSQ\cong\triangle RTS\); Reason: SSS
6. Statement: \(\angle RQS=\angle RTS\); Reason: CPCTC