Question

use the given information to prove that $\triangle rst \cong \triangle tur$.
given: $\overline{rs} \cong \overline{tu}$, $\angle srt \cong \angle utr$
prove: $\triangle rst \cong \triangle tur$

statement	reason
2 $\angle srt \cong \angle utr$	given
3 $\overline{rt} \cong \overline{rt}$	reason?
4 $\triangle rst \cong \triangle tur$	reason?

Explanation:

Step1: Analyze Statement 3

Statement 3 is \(\overline{RT} \cong \overline{RT}\). This is the reflexive property of congruence, which states that any segment is congruent to itself. So the reason for statement 3 is "Reflexive Property of Congruence".

Step2: Analyze Statement 4

We have \(\overline{RS} \cong \overline{TU}\) (given), \(\angle SRT \cong \angle UTR\) (given), and \(\overline{RT} \cong \overline{RT}\) (reflexive property). By the Side - Angle - Side (SAS) Congruence Postulate, if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. So for \(\triangle RST \cong \triangle TUR\), the reason is "SAS Congruence Postulate".

Answer:

• For Statement 3 ( \(\overline{RT} \cong \overline{RT}\) ), Reason: Reflexive Property of Congruence.
• For Statement 4 ( \(\triangle RST \cong \triangle TUR\) ), Reason: SAS Congruence Postulate.