Isosceles Triangle Proof B4Db42

1. **Problem Statement:** We are given that segment $CT$ bisects angle $SCA$ and that angle $S$ is congruent to angle $A$. We need to prove that triangle $SCA$ is isosceles with $SC \cong CA$. 2. **Key Definitions and Theorems:** - An angle bisector divides an angle into two equal angles. - If two angles in a triangle are congruent, then the sides opposite those angles are congruent (Isosceles Triangle Theorem). 3. **Step-by-step Proof:** - Since $CT$ bisects angle $SCA$, we have $$\angle SCT \cong \angle ACT.$$ This means the angle at vertex $C$ is split into two equal parts. - Given that $$\angle S \cong \angle A,$$ the angles at vertices $S$ and $A$ are equal. - In triangle $SCA$, two angles are congruent: $$\angle S = \angle A.$$ - By the Isosceles Triangle Theorem, if two angles of a triangle are congruent, then the sides opposite those angles are congruent. - The side opposite $$\angle S$$ is $$CA$$, and the side opposite $$\angle A$$ is $$SC$$. - Therefore, $$SC \cong CA$$. 4. **Conclusion:** Triangle $SCA$ has two congruent sides, $SC$ and $CA$, so it is an isosceles triangle.