Question

Triangle ABC is a right triangle. Point D is the midpoint of side AB and point E is the midpoint 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°. Which theorem can be used to fill in the missing reason?

Answer 1

The missing reason in the proof is likely related to the Angle Bisector Theorem or the properties of a right triangle, particularly the Midpoint Theorem and the fact that the sum of angles in any triangle is 180°.

Step-by-step explanation:

In the context of the given right triangle ABC, we have the midpoint D on side AB and midpoint E on side AC. Angle ADE measures 36°, and we need to prove that angle ECB measures 54°. The missing reason in the proof is likely related to the Angle Bisector Theorem or the properties of a right triangle and the fact that the sum of angles in any triangle is 180°. Since D and E are midpoints, we can infer that DE is parallel to BC because of the Midpoint Theorem, also known as the Triangle Midsegment Theorem, which states that the segment connecting the midpoints of two sides of a triangle is parallel to the third side and half its length. This implies that angle ADE is congruent to angle ACB because they are corresponding angles. Hence, if angle ADE is 36°, angle ACB is also 36°. Since ABC is a right triangle, angle B is 90°, and thus angle ACB (36°) and angle ABC must sum to 90°. Therefore, angle ABC must be 90° - 36° = 54°, making the measure of angle ECB equal to 54° as well.