Question

Equilateral triangle ABC. AD - its altitude. E - side's AB midpoint. What are the angles of the triangle AND?

Answer 1

Angle DEA measures 120°, angles EAD and ADE each measure 30°.

Explanation:

With the triangle properly labeled AED in the comments, we proceed as follows:

1) AD is an altitude so D lies opposite A on side BC. Since triangle ABC is equilateral then D is the midpoint of BC.

2) With D as the midpoint of one side BC and E as the _midpoint of a second side AB, line segment DE must be parallel to the third side AC. This means angle DEA is supplementary to angle BAC, and the latter angle in the equilateral triangle measures 60°. Therefore angle DEA measures 120°.

3) Side DE measures half the parallel side AC and EA measures half of AB. But sides AB and AC are congruent in the equilateral triangle, so DE and EA are congruent sides of an isosceles triangle AED. We just saw that the apex angle where these legs intersect measures 120°. So the two base angles, EAD and ADE, which are congruent must add up to 180°-120°=60°. So angles EAD and ADE each measure 30°.