Final answer:
To determine the measure of angle ECB, the Midsegment of a Triangle Theorem is applied to understand that DE is parallel to BC and thus angles ADE and DEB are congruent. Subsequently, the sum of the angles in a triangle and the properties of an isosceles triangle are used to find that angle ECB measures 54°.
Step-by-step explanation:
The student has encountered a geometry problem related to right triangles and the measure of angles. Given that triangle ABC is a right triangle with D being the midpoint of side AB and E the midpoint of AC, we know the measure of angle ADE is 36°. The task is to prove that angle ECB measures 54°.
To solve this, consider that DE is a midsegment in triangle ABC, connecting the midpoints of two sides. According to the Midsegment of a Triangle Theorem, DE is parallel to BC, and DE is half the length of BC. Since angle ADE is 36°, and DE is parallel to BC, angle DEB also measures 36° because they are alternate interior angles. Triangle ADE is therefore an isosceles triangle, where angles ADE and AED are equal. We know that angle BAC, the right angle, measures 90°, and angle ADE is 36°. Thus, the measure of angle AED is also 36°. The measure of angle DEC is 180° - 36° - 36° = 108°. Since DE is parallel to CB and angle DEC is an exterior angle to triangle CEB, it is equal to the sum of the remote interior angles, which means angle ECB must measure 54° (as 108° is double of 54°).