In triangle ABC the angle at B is 75° and the angle at C is 35°. From A both the altitude AD and the bisector AE are drawn, with D and E lying on BC. What is the measure of ∠DAE?

The angle at A is 180° − 75° − 35° = 70°, so the bisector gives ∠BAE = 35°. In the right triangle ABD, ∠BAD = 90° − 75° = 15°. Both D and E lie on BC and D is the nearer to B, so ∠DAE = ∠BAE − ∠BAD = 35° − 15° = 20°. This agrees with the general result ∠DAE = (∠B − ∠C)/2 = (75° − 35°)/2 = 20°. The traps 35° and 15° are the two angles that had to be subtracted, and 55° is 90° − 35°.