Exercise 5 Page 611

Let's first show all of the given information in the diagram: Note that $\angle QPS$ and $\angle PQR$ may look like right angles but we do not know for sure that they are. Therefore, we will not mark them as such.

To show that $△PTS\cong △QTR,$ we will first show that $△QPS\cong △PQR.$ Since these triangles share $\overline{PQ}$ as a side, we can claim by the Reflexive Property of Congruence that they are congruent. Let's separate these triangles to get a better understanding.

From the diagram we see that two sides and the included angle of $△PQS$ are congruent to two sides and the included angle of $△P^{\prime }{Q}^{\prime }S.$ Therefore, we can conclude by the SAS Congruence Theorem that they are congruent. We now know the following: Knowing that $\angle 1\cong \angle 2,$ we can write the following equations due to the Angle Addition Postulate: Since $\angle QPS\cong \angle PQR$ we can equate the left-hand sides of these equations. The measures of $m\angle SPT$ and $\angle RQT$ are the same which means we can also claim that $\angle SPT\cong \angle RQT.$ Let's merge the two triangles and add all of this information to the diagram.

Note that the base angles of $△PQT$ are congruent. According to the Converse of the Base Angles Theorem, if two angles of a triangle are congruent, then the sides opposite them are congruent.

From the diagram we see that two sides and the included angle of $△PTS$ are congruent with two sides and the included angle of $△QTR.$ Therefore, we can, by the SAS Congruence Theorem, conclude that the triangles are congruent.