Before you consider â–³ PTS and â–³ QTR, you might want to look at â–³ QPS and â–³ 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 â–³ PQS are congruent to two sides and the included angle of â–³ 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 â–³ 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 â–³ PTS are congruent with two sides and the included angle of â–³ QTR. Therefore, we can, by the SAS Congruence Theorem, conclude that the triangles are congruent.
