1. **Problem statement:** Given triangle ABC with angles $\hat{A} = 80^\circ$ and $\hat{B} = 50^\circ$, prove that triangle ABC is isosceles. 2. **Step 1: Find the third angle $\hat{C}$.** Using the triangle angle sum property: $$\hat{A} + \hat{B} + \hat{C} = 180^\circ$$ Substitute known values: $$80^\circ + 50^\circ + \hat{C} = 180^\circ$$ Simplify: $$\hat{C} = 180^\circ - 130^\circ = 50^\circ$$ 3. **Step 2: Analyze the angles.** We have $\hat{B} = 50^\circ$ and $\hat{C} = 50^\circ$. Since two angles are equal, the sides opposite those angles are equal by the Isosceles Triangle Theorem. 4. **Step 3: Conclusion.** Therefore, triangle ABC is isosceles with $AB = AC$ because $\hat{B} = \hat{C}$. Final answer: Triangle ABC is isosceles with $AB = AC$ because $\hat{B} = \hat{C} = 50^\circ$.