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 47°. the proof, with a missing reason, proves that the measure of ∠ecb is 43°.

the table info is below.

statement reason
m∠ade = 47° =given
m∠dae = 90° =definition of a right angle
m∠aed = 43° =?
segment de joins the midpoints of segment ab and segment ac =given
segment de is parallel to segment bc =midsegment of a triangle theorem
∠ecb ≅ ∠aed =corresponding angles are congruent
m∠ecb = 43° =substitution property

which theorem can be used to fill in the missing reason?
concurrency of medians theorem
isosceles triangle theorem
triangle inequality theorem
triangle sum theorem

Therefore The Missing reason is

Triangle Sum Theorem.

Step-by-step explanation:

Given:

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 47°.

To Prove

measure of ∠ECB is 43°.

Proof:

Triangle Sum Theorem :

The Triangle Sum Theorem states that the three interior angles of any triangle add up to 180 degrees.

∴ 90 + 47 + m∠AED = 180°

∴ m∠AED = 180 - 137 = 43°

Statement           Reason

m∠ADE = 47°    =   Given

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

m∠AED = 43°    =   Triangle Sum Theorem

segment DE joins the midpoints of segment AB and segment AC =Given

segment DE is parallel to segment BC =Midsegment of a Triangle Theorem

∠ECB ≅ ∠AED =Corresponding angles are congruent

m∠ECB = 43° =Substitution property

Therefore The Missing reason is

Triangle Sum Theorem

Below are the correct choices:

Corresponding angles are congruent

Triangle Sum Theorem

Measure of angle AED is 43°.

Alternate interior angles are congruent

Base Angle Theorem

Measure of angle AED is 47°

Base Angle Theorem

Corresponding angle are congruent

Measure of angle AED is 43°.

Alternate interior angles are congruent

Triangle Sum Theorem

Measure of angle AED is 47°

The answer is the 3rd one. I hope it helps. 

Step-by-step explanation: