Exercise 13 Page 364

By the definition of a parallelogram, we can say that AB∥DC. Definition of Parallelogram AB∥DC In this case, ∠ 1 and ∠ 4, and ∠2 and ∠3 are alternate interior angles. Therefore, we can conclude that they are congruent by the Alternate Interior Angles Theorem. Alternate Interior Angles Theorem ∠ 1 ≅ ∠ 4; ∠2 ≅ ∠3 By the Parallelogram Opposite Sides Theorem, we know that the opposite sides of a parallelogram are congruent. Parallelogram Opposite Sides Theorem AB ≅ DC 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. ASA Congruence Theorem △ ABE ≅ △ CDE Since corresponding parts of congruent triangles are congruent, we know that the other sides of the triangles are congruent as well. Definition of Congruent Triangles AE ≅ CE; BE ≅ DE Now, by the definition of a bisector, we can complete our proof. Definition of Bisector AC and BD bisect each other at E. By combining these steps, let's complete the two-column proof.