Question: Knowing that segment $CT$ bisects angle $SCA$ and that angle $S$ is congruent to angle $A$, write a paragraph proving that triangle $SCA$ is an isosceles triangle with segment $SC$ congruent to segment $CA$. Be sure to reference appropriate postulates, properties, or theorems from your Beloved Toolbox.

1. **State the problem:** We need to prove that triangle $SCA$ is isosceles, meaning $SC \cong CA$, given that segment $CT$ bisects angle $SCA$ and angle $S \cong$ angle $A$. 2. **Given information:** - $CT$ bisects angle $SCA$, so $\angle SCT \cong \angle ACT$ by definition of angle bisector. - $\angle S \cong \angle A$ (given). 3. **Goal:** Prove $SC \cong CA$. 4. **Use the Angle-Side-Angle (ASA) Postulate:** - In triangles $SCT$ and $ACT$: - $\angle SCT \cong \angle ACT$ (angle bisector) - $CT$ is common side (shared by both triangles) - $\angle S \cong \angle A$ (given) 5. **Apply ASA:** Triangles $SCT$ and $ACT$ are congruent by ASA postulate. 6. **Corresponding parts of congruent triangles are congruent (CPCTC):** - Therefore, $SC \cong CA$. 7. **Conclusion:** Since $SC \cong CA$, triangle $SCA$ has two congruent sides and is therefore isosceles. This proof uses the definition of angle bisector, the ASA postulate, and CPCTC from the Beloved Toolbox.