Answer
1) Use method SAS with the information given to prove $\triangle SPV\cong\triangle SQT$
2) First, deduce $\overline{PV}\cong\overline{QT}$ (from part 1).
Then, prove that $\overline{TP}\cong\overline{VQ}$
Finally, use method SSS to prove $\triangle TPQ\cong\triangle VQP$
Work Step by Step
1) Prove that $\triangle SPV\cong\triangle SQT$
Considering $\triangle SPV$ and $\triangle SQT$, we see that
- $\overline{SP}\cong\overline{SQ}$ (given)
- $\angle S\cong\angle S$ (by Identity)
- $\overline{SV}\cong\overline{ST}$ (given)
Therefore, by method SAS, we can conclude that $\triangle SPV\cong\triangle SQT$.
2) From the conclusion $\triangle SPV\cong\triangle SQT$, we can deduce that $\overline{PV}\cong\overline{QT}$.
*It is given that $\overline{SP}\cong\overline{SQ}$
So, $$\overline{ST}+\overline{TP}\cong\overline{SV}+\overline{VQ}$$
Yet, it is already given that $\overline{SV}\cong\overline{ST}$
Therefore, $\overline{TP}\cong\overline{VQ}$
*Considering $\triangle TPQ$ and $\triangle VQP$, we see that
- $\overline{TQ}\cong\overline{VP}$ (proved above)
- $\overline{TP}\cong\overline{VQ}$ (proved above)
- $\overline{PQ}\cong\overline{QP}$
So, according to method SSS, $\triangle TPQ\cong\triangle VQP$.
