1. **State the problem:** We are given a triangle with one interior angle measuring 65 degrees and its corresponding exterior angle measuring 115 degrees. We need to find the adjacent interior angle of the triangle at that vertex. 2. **Recall the properties:** The exterior angle of a triangle at a vertex is equal to the sum of the two opposite interior angles. 3. **Use the exterior angle theorem:** If the exterior angle is 115 degrees and one interior angle at that vertex is 65 degrees, then the adjacent interior angle $x$ satisfies: $$x + 65 = 180$$ because interior and exterior angles at the same vertex are supplementary. 4. **Solve for $x$:** $$x = 180 - 65 = 115$$ 5. **Interpretation:** The adjacent interior angle at that vertex is 115 degrees. **Final answer:** The adjacent interior angle of the triangle is $115$ degrees.