## Elementary Geometry for College Students (5th Edition)

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 $\triangle TPQ\cong\triangle VQP$
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, $\triangle TPQ\cong\triangle VQP$.