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 following flowchart with missing statements and reasons proves that the measure of angle ECB is 54°:



Which statement and reason can be used to fill in the numbered blank spaces?
1. Measure of angle AED is 36°
2. Base Angle Theorem
3. Corresponding angles are congruent
1. Measure of angle AED is 54°
2. Base Angle Theorem
3. Alternate interior angles are congruent
1. Measure of angle AED is 54°
2. Triangle Sum Theorem
3. Alternate interior angles are congruent

Answer 1

The statement and reason that can be used to fill the numbered blank spaces is the fourth option

1. Measure of angle AED is 54°
2. Triangle Sum Theorem
3. Corresponding angles are congruent

The statement and reasons are presented as follows;

Segment DE joins the midpoints of segments AB and AC; Given

Segment DE is parallel to segment BC; Midsegment Theorem

Measure of angle ADE is 36°; Given

Measure of angle DAE is 90°; Definition of right Angle

1. Measure of angle AED is 54°; 2. Triangle sum theorem

Angle ECB is congruent to angle AED; 3. Corresponding angles are congruent

Measure of angle ECB is 54°; Substitution property

The triangle sum theorem states that the sum of the interior angles in a triangle is 180°, therefore; m∠AED + m∠ADE + m∠DAE = 180°

m∠AED + = 180°

m∠AED = 180° - (36° + 90°)

m∠AED = 54°

The corresponding angles theorem states that the corresponding angles formed between parallel lines and their transversal are congruent

Angle ∠ECB and angle ∠AED are corresponding angles, therefore;

Angle ECB is congruent to angle AED

m∠ECB = m∠AED (Definition of congruent angles)

The substitution property states that if A = B then B can substitute A in a specified equation and the equation remain valid

m∠AED = 54° and m∠ECB = m∠AED, therefore, m∠ECB = 54°

Please find attached the possible flowchart created with MS Word, obtained from a similar question found through search