Question

proof complete the paragraph proof. given: \\(\overline{qr} \cong \overline{sr}\\), \\(\overline{st} \cong \overline{qt}\\) prove: \\(\triangle qrt \cong \triangle srt\\) proof: we know that \\(\overline{qr} \cong \overline{sr}\\) and \\(\overline{st} \cong \overline{qt}\\). \\(\overline{rt} \cong \overline{rt}\\) by the \\(\boldsymbol{\text{select choice}}\\) property. because \\(\overline{qr} \cong \overline{sr}\\), \\(\overline{st} \cong \overline{qt}\\), and \\(\overline{rt} \cong \overline{rt}\\), \\(\triangle qrt \cong \triangle srt\\) by \\(\boldsymbol{\text{select choice}}\\).

Explanation:

Step1: Identify RT's property

RT is a side common to both $\triangle QRT$ and $\triangle SRT$. A common side is congruent to itself by the Reflexive Property of Congruence. So $RT \cong RT$ by Reflexive Property.

Step2: Identify triangle congruence criterion

We have $QR \cong SR$, $ST \cong QT$, and $RT \cong RT$. This matches the SSS (Side - Side - Side) Congruence Criterion, which states that if three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent. So $\triangle QRT \cong \triangle SRT$ by SSS.

Answer:

First blank: Reflexive; Second blank: SSS (Side - Side - Side) Congruence Criterion