Question

△ruv≅△stv. complete the proof that △rsu≅△srt.
1 △ruv≅△stv given
2 \overline{ru}≅\overline{st} cpctc
3 \overline{rv}≅\overline{sv}
4 \overline{tv}≅\overline{uv} cpctc
5 ∠ruv≅∠stv
6 rt = rv + tv additive property of length
7 su = sv + uv additive property of length
8 rt = sv + uv substitution
9 rt = su transitive property of equality
10 △rsu≅△srt sas

Explanation:

Step1: Identify the reason for $\overline{RV}\cong\overline{SV}$

Since $\triangle RUV\cong\triangle STV$, by CPCTC (Corresponding Parts of Congruent Triangles are Congruent), $\overline{RV}\cong\overline{SV}$.

Step2: Identify the reason for $\angle RUV\cong\angle STV$

Again, because $\triangle RUV\cong\triangle STV$, by CPCTC, $\angle RUV\cong\angle STV$.

Step3: Verify the SSS (Side - Side - Side) or SAS (Side - Angle - Side) for $\triangle RSU\cong\triangle SRT$

We have $\overline{RU}\cong\overline{ST}$ (from step 2 of the given proof, CPCTC), $\overline{SU}\cong\overline{RT}$ (from step 9, $RT = SU$), and $\overline{RS}\cong\overline{SR}$ (common side). But the proof mentions "SAS".
Let's check the angle. We know $\angle RUV+\angle RUS = 180^{\circ}$ and $\angle STV+\angle STR=180^{\circ}$ (linear pairs). Since $\angle RUV\cong\angle STV$, then $\angle RUS\cong\angle STR$.
Also, $\overline{RU}\cong\overline{ST}$, $\overline{SU}\cong\overline{RT}$. But if we use the SAS criterion:
We have $\overline{RU}\cong\overline{ST}$, $\angle RUS\cong\angle STR$, $\overline{SU}\cong\overline{RT}$.

Answer:

The reason for $\overline{RV}\cong\overline{SV}$ is CPCTC. The reason for $\angle RUV\cong\angle STV$ is CPCTC. For the congruence $\triangle RSU\cong\triangle SRT$ using SAS: we have $\overline{RU}\cong\overline{ST}$ (from $\triangle RUV\cong\triangle STV$, CPCTC), $\angle RUS\cong\angle STR$ (supplements of congruent angles $\angle RUV$ and $\angle STV$ are congruent), and $\overline{SU}\cong\overline{RT}$ (from $RT = SU$). So $\triangle RSU\cong\triangle SRT$ by SAS.