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°. Triangle ABC with segment DE. Angle ADE measures 36 degrees. 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 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 = 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

Answer 1

Answer:

The correct options is;

Midsegment of a Triangle Theorem

Step-by-step explanation:

The Midsegment of a Triangle Theorem states that when there is a segment that joins the midpoints of two adjacent sides of a triangle, that segment will be half the length of the third side of the triangle and parallel to it (the third side of the triangle)

Therefore, each triangle can have at least, three midsegments constructed to be parallel to each of the three sides

m∠ADE = 36° (Given)

m∠DAE = 90°  (Definition of a right angle)

m∠AED = 54° (Triangle Sum Theorem)

Segment DE joins the midpoints of segments AB and AC (Given)

Segment DE is parallel to segment BC (Midsegment of a Triangle Theorem)

∠ECB ≅ ∠AED (Corresponding angles are congruent)

∴ ∠ECB = 54° (Substitution property).