Statement Reasons
PQ || ST; PR = RT | Given
∠RTS ≅ ∠RPQ | Alternate Interior Angles Theorem (TP is a transversal)
∠TSR ≅ ∠RQP | Alternate Interior Angles Theorem (SQ is a transversal)
∆PQR ≅ ∆TSR | SAS (Angle-Side-Angle) Congruence.
The question requires us to prove that triangles APQR and ATSR are congruent given that line segment PQ is parallel to line segment ST and that PR is congruent to RT. To methodologically approach this proof we will use the properties of parallel lines, congruent segments, and congruent angles.
PQ || ST; PR = RT - Given.
∠RTS ∠RPQ - Alternate Interior Angles Theorem (since TP is a transversal).
∠TSR ∠RQP - Alternate Interior Angles Theorem (since SQ is a transversal).
∠PQR ∠TSR - Vertical Angles Theorem.
Traingle APQR ≅ Traingle ATSR - ASA (Angle-Side-Angle) since we have two angles and the included side between them that are congruent.
The provided statements and reasons give us enough information to determine that the triangles are congruent by ASA, which means that if two angles and the included side of one triangle are equal to two angles and the included side of another triangle, then the triangles are congruent.