1. **State the problem:** We have trapezoid ABCD with legs BC and AD, DE perpendicular to CB, and given angles $m\angle A = 86^\circ$ and $m\angle C = 74^\circ$. We want to find the measures of $m\angle B$ and $m\angle ADE$. 2. **Recall trapezoid angle properties:** In trapezoid ABCD, the sum of the interior angles is $360^\circ$. Also, consecutive angles between parallel sides are supplementary. 3. **Find $m\angle B$:** Since ABCD is a trapezoid, angles $A$ and $B$ are consecutive angles on the same leg AD. We use the fact that $m\angle A + m\angle B = 180^\circ$ (because AD is a leg and AB is parallel to DC). Calculate: $$m\angle B = 180^\circ - m\angle A = 180^\circ - 86^\circ = 94^\circ$$ 4. **Find $m\angle D$:** Similarly, angles $C$ and $D$ are consecutive angles on leg BC. They satisfy: $$m\angle C + m\angle D = 180^\circ$$ Calculate: $$m\angle D = 180^\circ - 74^\circ = 106^\circ$$ 5. **Find $m\angle ADE$:** Since $DE \perp CB$, $m\angle DEB = 90^\circ$. Point E lies on segment CB, so triangle ADE is formed with $DE \perp CB$. In triangle ADE, angles sum to $180^\circ$: $$m\angle ADE + m\angle DEA + m\angle EAD = 180^\circ$$ We know $m\angle DEA = 90^\circ$ (right angle at E). Angle $m\angle EAD$ is part of $m\angle A$, so: $$m\angle EAD = m\angle A = 86^\circ$$ Calculate $m\angle ADE$: $$m\angle ADE = 180^\circ - 90^\circ - 86^\circ = 4^\circ$$ **Final answers:** $$m\angle B = 94^\circ$$ $$m\angle ADE = 4^\circ$$