1. **Stating the problem:** We have an isosceles triangle with two equal base angles each measuring $50^\circ$. We need to find the apex angle, which is the angle opposite the base. 2. **Formula and rules:** The sum of the interior angles in any triangle is always $180^\circ$. For an isosceles triangle, the two base angles are equal. 3. **Calculation:** Let the apex angle be $x$. Since the two base angles are each $50^\circ$, the sum of the angles is: $$x + 50 + 50 = 180$$ 4. **Simplify:** $$x + 100 = 180$$ 5. **Solve for $x$:** $$x = 180 - 100 = 80$$ 6. **Conclusion:** The apex angle measures $80^\circ$. This means the triangle has angles $50^\circ$, $50^\circ$, and $80^\circ$.