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 68°.

The following flowchart with missing statements and reasons proves that the measure of angle ECB is 22°:

Which statement and reason can be used to fill in the numbered blank spaces?

Corresponding angles are congruent
Triangle Sum Theorem
Measure of angle AED is 22°

Corresponding angles are congruent
Base angle theorem
Measure of angle AED is 68°

Alternate interior angles are congruent
Triangle Sum Theorem
Measure of angle AED is 22°

Alternate interior angles are congruent
Triangle angle sum theorem
Measure of angle AED is 68°

Answer 1

Given that,

"Triangle ABC is a right triangle.

Point D is the midpoint of side AB

Point E is the midpoint of side AC.

The measure of angle ADE is 68°."

Angle DAE is $90^\circ$

Using the Triangle Sum Theorem of triangle ADE,

Angle ADE +Angle DAE +Angle DEA = 180°

Angle DEA = 180° -68°-90°= 22°

Again that Segment DE joins the midpoint of AB and AC.

DE is parallel to BC.

Thus Angle ECB is congruent to the angle AED.

So Angle AED = 22°= ECB.

Thus the measure of angle ECB is 22°.

Answer 2

Since DE || BC, and AC is a transversal,

∠AED and ∠ECB are congruent.

In ΔADE, since two angles are given, the third angle can be found by using Triangle Sum Theorem.

So, ∠AED = 180° - (68° + 90°)

= 180° - 158°

= 22°

Hence, the correct statements and reasons are:

Corresponding angles are congruent

Triangle Sum Theorem

Measure of angle AED is 22°