1. **State the problem:** We have an isosceles triangle with two base angles each labeled $a^\circ$ and the vertex angle labeled $x^\circ$. The two sides are extended upwards forming an X shape, and the angle formed at the intersection above the triangle is also $x^\circ$. We need to find the relationship between $a$ and $x$. 2. **Recall the triangle angle sum rule:** The sum of interior angles in any triangle is $180^\circ$. For the isosceles triangle, $$a + a + x = 180$$ which simplifies to $$2a + x = 180$$ 3. **Use the exterior angle property:** The angle formed at the intersection above the triangle (outside) is also $x^\circ$. This angle is an exterior angle to the triangle formed by extending the sides. 4. **Exterior angle theorem:** An exterior angle of a triangle equals the sum of the two opposite interior angles. Here, the exterior angle $x$ equals the sum of the two base angles $a + a$: $$x = 2a$$ 5. **Combine the two equations:** From step 2, we have $$2a + x = 180$$ Substitute $x = 2a$ from step 4: $$2a + 2a = 180$$ $$4a = 180$$ $$a = 45$$ 6. **Find $x$:** Using $x = 2a$: $$x = 2 \times 45 = 90$$ **Final answer:** $$a = 45^\circ, \quad x = 90^\circ$$