Exercise 13 Page 364

To prove the given statement, we will write a two-column proof.

To write a proof, we always begin by stating the given information. By the definition of a parallelogram, we can say that $\overline{AB}\parallel \overline{DC}.$ In this case, $\angle 1$ and $\angle 4,$ and $\angle 2$ and $\angle 3$ are alternate interior angles. Therefore, we can conclude that they are congruent by the Alternate Interior Angles Theorem. By the Parallelogram Opposite Sides Theorem, we know that the opposite sides of a parallelogram are congruent. Combining all of the previous steps, we have shown that two angles and their included side of $△ABE$ are congruent to two angles and their included side of $△CDE.$ Therefore, by the Angle-Side-Angle Congruence Theorem, we can write a congruence statement relating the two triangles. Since corresponding parts of congruent triangles are congruent, we know that the other sides of the triangles are congruent as well. Now, by the definition of a bisector, we can complete our proof. By combining these steps, let's complete the two-column proof.

Statement heading	Reason heading
$ABCD$ is a parallelogram.	Given
$\overline{AB}\parallel \overline{DC}$	a. Definition of Parallelogram
$\angle 1\cong \angle 4;$ $\angle 2\cong \angle 3$	b. Alternate Interior Angles Theorem
$\overline{AB}\cong \overline{DC}$	c. Parallelogram Opposite Sides Theorem
d. $△ABE\cong △CDE$	Angle-Side-Angle Congruence Theorem
$\overline{AE}\cong \overline{CE};$ $\overline{BE}\cong \overline{DE}$	e. Definition of Congruent Triangles
$\overline{AC}$ and $\overline{BD}$ bisects each other at $E.$	Definition of Bisector