1. **Problem 1:** Given $BD \perp AB$, $BD \perp DE$, and $BC \cong DC$, prove $\angle A \cong \angle E$. 2. **Problem 2:** Given $BC \cong DC$, $AC \cong EC$, prove $\triangle ABC \cong \triangle EDC$. 3. **Problem 3:** Given $YA \cong BA$, $\angle B \cong \angle Y$, prove $AZ \cong AC$. --- ### Step-by-step solutions: ### Problem 1: 1. We know $BD$ is perpendicular to both $AB$ and $DE$, so $\angle BDA$ and $\angle BDE$ are right angles. 2. Since $BC \cong DC$ (given), triangles $BCD$ and $DCE$ share side $BD$ and have two right angles. 3. By the Hypotenuse-Leg (HL) theorem for right triangles, $\triangle BCD \cong \triangle DCE$. 4. Corresponding parts of congruent triangles are congruent (CPCTC), so $\angle A \cong \angle E$. ### Problem 2: 1. Given $BC \cong DC$ and $AC \cong EC$. 2. Side $BC$ corresponds to side $DC$, and side $AC$ corresponds to side $EC$. 3. Side $BC$ and $DC$ meet at vertex $C$, and $AC$ and $EC$ also meet at $C$. 4. By the Side-Side-Side (SSS) congruence postulate, $\triangle ABC \cong \triangle EDC$. ### Problem 3: 1. Given $YA \cong BA$ and $\angle B \cong \angle Y$. 2. $\angle YAZ$ and $\angle BAC$ are vertical angles, so $\angle YAZ \cong \angle BAC$. 3. Triangles $YAZ$ and $BAC$ have two angles and the included side congruent (ASA). 4. By ASA postulate, $\triangle YAZ \cong \triangle BAC$. 5. By CPCTC, $AZ \cong AC$. --- ### Summary of used congruence postulates: - HL (Hypotenuse-Leg) for right triangles. - SSS (Side-Side-Side). - ASA (Angle-Side-Angle). These postulates allow us to prove triangle congruence and corresponding angle or side congruence.