1. **State the problem:** We need to find the measure of angle 1 ($m\angle 1$) in a triangle where one interior angle is $40^\circ$ and an exterior angle adjacent to the base is $138^\circ$. 2. **Recall the exterior angle theorem:** The measure of an exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles. 3. **Apply the theorem:** Given the exterior angle is $138^\circ$, and one interior angle is $40^\circ$, let the other interior angle adjacent to angle 1 be $x$. Then: $$138 = 40 + x$$ 4. **Solve for $x$:** $$x = 138 - 40 = 98^\circ$$ 5. **Use the triangle angle sum property:** The sum of interior angles in a triangle is $180^\circ$. $$m\angle 1 + 40 + 98 = 180$$ 6. **Calculate $m\angle 1$:** $$m\angle 1 = 180 - 40 - 98 = 42^\circ$$ **Final answer:** $$m\angle 1 = 42^\circ$$