Prove ΔFAT ≅ ΔOST with Segment Bisector and Angle Congruence Given

Question

Given: $\overline{FO}$ bisects $\overline{AS}$, $\angle A \cong \angle S$. Prove: $\Delta FAT \cong \Delta OST$

Accepted Answer

$\Delta FAT \cong \Delta OST$ is proven using AAS congruence criterion

Answer

Step 1: Identify the segment bisector result. Since $\overline{FO}$ bisects $\overline{AS}$, by definition of a segment bisector, it divides $\overline{AS}$ into two equal parts, so $\overline{AT} \cong \overline{ST}$. Step 2: Identify vertical angles. $\angle FTA$ and $\angle OTS$ are vertical angles formed by intersecting lines $\overline{FO}$ and $\overline{AS}$, so $\angle FTA \cong \angle OTS$ (vertical angles theorem). Step 3: List given congruent angles. We are given $\angle A \cong \angle S$. Step 4: Apply AAS congruence criterion. In $\Delta FAT$ and $\Delta OST$, we have $\angle A \cong \angle S$, $\angle FTA \cong \angle OTS$, and $\overline{AT} \cong \overline{ST}$. By the Angle-Angle-Side (AAS) triangle congruence postulate, $\Delta FAT \cong \Delta OST$.