Question

r is the midpoint of \\( \overline{qs} \\), \\( \overline{rt} \cong \overline{su} \\), and \\( \overline{qt} \cong \overline{ru} \\). complete the proof that \\( \triangle qrt \cong \triangle rsu \\).

Explanation:

Step1: Given

The problem statement provides the information.

Step2: Given

The problem statement provides the information.

Step3: Given

The problem statement provides the information.

Step4: Definition of mid - point

If \(R\) is the mid - point of \(\overline{QS}\), then \(QR = RS\), so \(\overline{QR}\cong\overline{RS}\)

Step5: SSS (Side - Side - Side) Congruence Criterion

In \(\triangle QRT\) and \(\triangle RSU\), we have \(\overline{QR}\cong\overline{RS}\), \(\overline{RT}\cong\overline{SU}\), \(\overline{QT}\cong\overline{RU}\). By SSS, \(\triangle QRT\cong\triangle RSU\)

Answer:

Statement	Reason
2. \(\overline{RT}\cong\overline{SU}\)	Given
3. \(\overline{QT}\cong\overline{RU}\)	Given
4. \(\overline{QR}\cong\overline{RS}\)	Definition of mid - point
5. \(\triangle QRT\cong\triangle RSU\)	SSS (Side - Side - Side) Congruence Criterion