Question

$\overline{rt} \cong \overline{su}$. complete the proof that $\overline{ru} \cong \overline{st}$.

statement	reason
2 $\overline{rt} \cong \overline{su}$	given
3 $\angle srt \cong \angle rsu$	given
4 $\angle srt \cong \angle rtu$	alternate interior angles theorem
5 $\angle sut \cong \angle rsu$	alternate interior angles theorem
6 $\angle rsu \cong \angle rtu$	transitive property of congruence
7 $\angle sut \cong \angle rtu$	transitive property of congruence
8 $\overline{tu} \cong \overline{tu}$	reflexive property of congruence
9

Explanation:

Step1: Identify Triangle Congruence

We have \( \overline{RT} \cong \overline{SU} \) (given), \( \overline{TU} \cong \overline{TU} \) (reflexive), and \( \angle SUT \cong \angle RTU \) (from step 7). So by ASA (Angle - Side - Angle) congruence criterion, \( \triangle SUT \cong \triangle RTU \).

Step2: Corresponding Parts of Congruent Triangles

Since \( \triangle SUT \cong \triangle RTU \), their corresponding sides \( \overline{RU} \) and \( \overline{ST} \) are congruent (CPCTC - Corresponding Parts of Congruent Triangles are Congruent). So the statement for step 9 is \( \overline{RU} \cong \overline{ST} \) with the reason "CPCTC (Corresponding Parts of Congruent Triangles are Congruent)".

Answer:

Statement: \( \overline{RU} \cong \overline{ST} \); Reason: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)