Before you consider △ PTS and △ QTR, you might want to look at △ QPS and △ PQR.
See solution.
Practice makes perfect
Let's first show all of the given information in the diagram:
PS≅ QR and ∠ QPS ≅ ∠ PQR
Note that ∠ QPS and ∠ 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≅ △ QTR, we will first show that △ QPS≅ △ PQR. Since these triangles share 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'Q'S. Therefore, we can conclude by the SAS Congruence Theorem that they are congruent. We now know the following:
∠ 1≅ ∠ 2 and ∠ 3 ≅ ∠ 4.
Knowing that ∠ 1 ≅ ∠ 2, we can write the following equations due to the Angle Addition Postulate:
&m∠ 1+ m∠ SPT=m∠ QPS
&m∠ 2+ m∠ RQT=m∠ PQR
Since ∠ QPS ≅ ∠ PQR we can equate the left-hand sides of these equations.
The measures of m∠ SPT and ∠ RQT are the same which means we can also claim that ∠ SPT≅ ∠ 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.
