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 47°.
Triangle ABC with segment DE. Angle ADE measures 47 degrees.

The following flowchart with missing statements and reasons proves that the measure of angle ECB is 43°:

Statement, Measure of angle ADE is 47 degrees, Reason, Given, and Statement, Measure of angle DAE is 90 degrees, Reason, Definition of right angle, leading to Statement 3 and Reason 2, which further leads to Statement, Measure of angle ECB is 43 degrees, Reason, Substitution Property. Statement, Segment DE joins the midpoints of segment AB and AC, Reason, Given, leading to Statement, Segment DE is parallel to segment BC, Reason, Midsegment theorem, which leads to Angle ECB is congruent to angle AED, Reason 1, which further leads to Statement, Measure of angle ECB is 43 degrees, Reason, Substitution Property.

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 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°

Answer 1

Correct Answer:

1. Corresponding angles are congruent

2. Triangle Sum Theorem

3. Measure of angle AED is 43°

Answer 2

Answer:

Step-by-step explanation:

It is given that 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 47°.

To prove: The measure of angle ECB is 43°.

Proof:

Statement                                                          Reason

1. m∠ADE=47°                                                     Given

2. m∠DAE=90°                                        definition of right angle

3. In ΔADE, ∠ADE+∠DAE+∠DEA=180°      

⇒47+90+∠DEA=180°

⇒∠DEA=43°                                              triangle sum theorem

4. ∠ECB=43°                                              substitution property

5.Segment DE joins the midpoints

of segment AB and AC                                        Given

6. Segment DE is parallel to segment BC        Midsegment theorem

7. m∠ECB≅m∠AED                                Corresponding angles are congruent

therefore, Measure of angle AED is 43°.

Hence proved.

Hence, option A is correct.