Prove ΔPQR ≅ ΔTSR with Midpoint and Angle Congruence Given

Question

Given: $\angle P \cong \angle T$, $R$ is the midpoint of $\overline{QS}$. Prove: $\Delta PQR \cong \Delta TSR$

Accepted Answer

$\Delta PQR \cong \Delta TSR$ is proven using AAS congruence criterion

Answer

Step 1: Identify midpoint result. Since $R$ is the midpoint of $\overline{QS}$, by definition of a midpoint, it divides $\overline{QS}$ into two equal parts, so $\overline{QR} \cong \overline{SR}$. Step 2: Identify vertical angles. $\angle PRQ$ and $\angle TRS$ are vertical angles formed by intersecting lines $\overline{PT}$ and $\overline{QS}$, so $\angle PRQ \cong \angle TRS$ (vertical angles theorem). Step 3: List given congruent angles. We are given $\angle P \cong \angle T$. Step 4: Apply AAS congruence criterion. In $\Delta PQR$ and $\Delta TSR$, we have $\angle P \cong \angle T$, $\angle PRQ \cong \angle TRS$, and $\overline{QR} \cong \overline{SR}$. By the Angle-Angle-Side (AAS) triangle congruence postulate, $\Delta PQR \cong \Delta TSR$.