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 ∠ADE is 68°. Triangle ABC with segment DE. Angle ADE measures 68 degrees. The proof, with a missing reason, proves that the measure of ∠ECB is 22°.

Statement | Reason
-----------------------------------|------------------------------------
m∠ADE = 68° | Given
m∠DAE = 90° | Definition of a right angle
m∠AED = 22° | Triangle sum theorem
Segment DE joins the |
midpoints of segment AB |
and segment AC |
Given segment DE is |
parallel to segment BC |
∠ECB ≅ ∠AED | Corresponding angles are congruent
m∠ECB = 22° | Substitution property

Which of the following can be used to fill in the missing reason?

Answer 1

The missing reason in the proof to establish that ∠ECB measures 22° relates to the Midpoint Theorem which states that the segment joining midpoints in a triangle is parallel to the third side.

Step-by-step explanation:

The missing reason in the proof that proves the measure of ∠ECB is 22° can be filled with the theorem stating that a line segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length.

This is known as the Midpoint Theorem, sometimes also referred to as the Triangle Midsegment Theorem. Since DE joins the midpoints of AB and AC and is given to be parallel to BC, the corresponding angles ∠AED and ∠ECB are congruent according to the corresponding angles postulate.