Prove the congruence of triangles △ PVT and △ SVQ first.
See solution.
Practice makes perfect
To create our flow proof, let's first investigate the relationships between different pairs of triangles.
Focusing on Triangles △ TPM and △ QSR
There are several markings on the diagram connecting triangles △ TPM and △ QSR.
Let's summarize some of the given congruences.
PM&≅SR
∠ M&≅∠ R
∠ P&≅∠ S
We can see that, in triangles △ TPM and △ QSR, two pairs of angles and the included sides are congruent.
According to the Angle-Side-Angle (ASA) Congruence Postulate, this implies that the triangles are congruent.
△ TPM≅△ QSR
Since corresponding parts of congruent triangles are congruent, this implies the congruence of other angles and segments.
One such congruence is particularly relevant in investigating the quadrilateral PQST in the next step.
TP≅QS
Focusing on Triangles △ PVT and △ SVQ
From above, we know that segments TP and QS are congruent.
From the markings, we also know that they are parallel.
Let's see what we know about the angles of these two triangles.
Since PS is a transversal to two parallel lines, the alternate interior angles ∠ TPS and ∠ QSP are congruent.
Since TQ is also a transversal to two parallel lines, the alternate interior angles ∠ QTP and ∠ TQS are also congruent.
We can see that triangles △ PVT and △ SVQ have two pairs of congruent angles and congruent included sides.
According to the Angle-Side-Angle (ASA) Congruence Postulate, this implies that the triangles are congruent.
△ PVT≅△ SVQ
Since corresponding parts of congruent triangles are congruent, this implies the congruence of other angles and segments.
There are two congruences which are relevant in investigating triangles △ PVQ and △ SVT in the next step.
PV&≅SV
VT&≅VQ
Focusing on Triangles △ PVQ and △ SVT
Let's mark the congruences we found previously on the diagram.
Notice that the angles ∠ PVQ and ∠ SVT are vertical angles. Hence, they are congruent.
In triangles △ PVQ and △ SVT, there are two pairs of congruent sides and their included angles are also congruent.
According to the Side-Angle-Side (SAS) Congruence Postulate, this implies that the triangles are congruent.
△ PVQ≅△ SVT
