1. **Problem Statement:** Prove that triangles $\triangle ABC$ and $\triangle FED$ are congruent, and determine if triangles $\triangle SRT$ and $\triangle DEF$ are similar, providing reasons. 2. **Given Data for $\triangle ABC$ and $\triangle FED$:** - $\angle A = 70^\circ$, $\angle C = 50^\circ$, side $BC = 10$ cm. - $\angle F = 70^\circ$, $\angle E = 60^\circ$, side $ED = 10$ cm. 3. **Step 1: Find the missing angles in $\triangle ABC$ and $\triangle FED$** - For $\triangle ABC$, sum of angles is $180^\circ$: $$\angle B = 180^\circ - 70^\circ - 50^\circ = 60^\circ$$ - For $\triangle FED$, sum of angles is $180^\circ$: $$\angle D = 180^\circ - 70^\circ - 60^\circ = 50^\circ$$ 4. **Step 2: Compare angles and sides** - $\triangle ABC$ angles: $70^\circ, 60^\circ, 50^\circ$ - $\triangle FED$ angles: $70^\circ, 60^\circ, 50^\circ$ - Side $BC = 10$ cm corresponds to side $ED = 10$ cm. 5. **Step 3: Use Angle-Side-Angle (ASA) Congruence Criterion** - Two angles and the included side in $\triangle ABC$ are equal to two angles and the included side in $\triangle FED$: - $\angle A = \angle F = 70^\circ$ - Side $BC = ED = 10$ cm - $\angle C = \angle D = 50^\circ$ - Therefore, $\triangle ABC \cong \triangle FED$ by ASA. 6. **Step 4: Determine similarity of $\triangle SRT$ and $\triangle DEF$** - $\triangle SRT$ angles: $\angle R = 75^\circ$, $\angle T = 45^\circ$, so $\angle S = 180^\circ - 75^\circ - 45^\circ = 60^\circ$ - $\triangle DEF$ angles: $\angle D = 75^\circ$, $\angle F = 60^\circ$, so $\angle E = 180^\circ - 75^\circ - 60^\circ = 45^\circ$ 7. **Step 5: Compare angles for similarity** - $\triangle SRT$ angles: $75^\circ, 60^\circ, 45^\circ$ - $\triangle DEF$ angles: $75^\circ, 60^\circ, 45^\circ$ - Since all corresponding angles are equal, $\triangle SRT \sim \triangle DEF$ by Angle-Angle (AA) similarity criterion. **Final Answers:** - $\triangle ABC \cong \triangle FED$ by ASA congruence. - $\triangle SRT \sim \triangle DEF$ by AA similarity.