Question

triangle abc is a right triangle. point d is the mid - point of side ab and point e is the mid - point of side ac. the measure of angle ade is 36°. the proof, with a missing reason, proves that the measure of angle ecb is 54°. statement reason m∠ade = 36° given m∠dae = 90° definition of a right angle m∠aed = 54° triangle sum theorem de joins the midpoints of ab and ac given de is parallel to bc? ∠ecb ≅ ∠aed corresponding angles are congruent m∠ecb = 54° substitution property which theorem can be used to fill in the missing reason? concurrency of medians theorem isosceles triangle theorem midsegment of a triangle theorem triangle inequality theorem

Explanation:

The Mid - segment of a Triangle Theorem states that a line segment joining the mid - points of two sides of a triangle is parallel to the third side. Here, since D is the mid - point of AB and E is the mid - point of AC, DE is parallel to BC by the Mid - segment of a Triangle Theorem. This justifies the step that DE is parallel to BC.

Answer:

Mid - segment of a Triangle Theorem