1. **Stating the problem:** We need to find the values of angles $h$ and $j$ in the given circle diagram where two chords intersect inside the circle. 2. **Relevant formula:** When two chords intersect inside a circle, the measure of each angle formed is half the sum of the measures of the arcs intercepted by the angle and its vertical angle. Mathematically, if two chords intersect at point $i$, then: $$\text{angle} = \frac{1}{2} (\text{arc}_1 + \text{arc}_2)$$ 3. **Given information:** One angle adjacent to $h$ is $39^\circ$. The chords intersect at $i$, forming vertical angles $h$ and $j$. 4. **Step to find $h$:** Since $h$ and the $39^\circ$ angle are vertical angles, they are equal. Therefore, $$h = 39^\circ$$ 5. **Step to find $j$:** Angles $h$ and $j$ are supplementary because they form a linear pair at the intersection. So, $$h + j = 180^\circ$$ Substitute $h = 39^\circ$: $$39^\circ + j = 180^\circ$$ 6. **Solve for $j$:** $$j = 180^\circ - 39^\circ = 141^\circ$$ **Final answers:** $$h = 39^\circ$$ $$j = 141^\circ$$