In triangle ABC, $\angle B = 74^\circ$ and $\angle C = 36^\circ$. D is the point on BC with AD perpendicular to BC, and E is the point on BC where the bisector of $\angle A$ meets it. What is $\angle DAE$?

Explanation

1. 01

   $\angle A = 180^\circ - 74^\circ - 36^\circ = 70^\circ$, so $\angle BAE = 35^\circ$.

2. 02

   In right triangle ABD, $\angle BAD = 90^\circ - 74^\circ = 16^\circ$.

3. 03

   D lies between B and E, so $\angle DAE = \angle BAE - \angle BAD = 35^\circ - 16^\circ = 19^\circ$.

4. 04

   Shortcut: $\angle DAE = \tfrac{1}{2}(\angle B - \angle C) = \tfrac{1}{2}(74^\circ - 36^\circ) = 19^\circ$.