Exercise 15 Page 337

We will complete the given proof by filling in the blanks. Let's start with examining the given figure and information and the desired conclusion.

We will start with blank a.

Blank a.

In order to fill in blank a., we will benefit from the given information $m\angle EAC=m\angle AEC.$ In this case, we will use the Converse of the Isosceles Triangle Theorem.

Converse of the Isosceles Triangle Theorem

If two angles of a triangle are congruent, then the sides opposite those angles are congruent.

Thus, we can complete the blank.

Blanks b., c. and d.

The statement in the third step is already given. For the fourth step, we will use the previous statement. Since the point $C$ is the midpoint, it divides $\overline{BD}$ into two equal parts, which is the definition of the midpoint. Since the congruent segments have equal lengths, we can immediately complete step five.

Blanks e. and f.

As we can see, the statement in step six is already given. For the last step, we will use the Hinge Theorem according to the sixth step.

Hinge Theorem

If two sides of one triangle are congruent to two sides of another triangle, and the included angles are not congruent, then the longer third side is opposite the larger included angle.

An illustration of the theorem is given below.

Applying the theorem to the given diagram, we can finish the proof.

Completed Proof

0. Statements	0. Reasons
1. $m\angle EAC=m\angle AEC$	1. Given
2. $AC=EC$	2. a. Converse of the Isosceles Triangle Theorem
3. $C$ is the midpoint of $\overline{BD}.$	3. b. Given
4. $\overline{BC}\cong \overline{CD}$	4. c. Definition of Midpoint
5. d. $\underline{BC=CD}$	5. $\cong$ segments have $=$ length.
6. $m\angle BCA>m\angle DCE$	6. e. Given
7. $AB>ED$	7. f. Hinge Theorem