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

Given : Prove : A B C D is a parallelogram . \overline{A C} and \overline{B D} bisect each other at E .

To write a proof, we always begin by stating the given information.

\underline{Given} A B C D is a parallelogram .

By the definition of a parallelogram, we can say that

\overline{A B} ∥ \overline{D C} .

\underline{Definition of Parallelogram} \overline{A B} ∥ \overline{D C}

In this case,

∠ 1

and

∠ 4,

and

∠ 2

and

∠ 3

\underline{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.

\underline{Parallelogram Opposite Sides Theorem} \overline{A B} ≅ \overline{D C}

Combining all of the previous steps, we have shown that two angles and their included side of

△ A B E

are congruent to two angles and their included side of

△ C D E .

Therefore, by the Angle-Side-Angle Congruence Theorem, we can write a congruence statement relating the two triangles.

\underline{ASA Congruence Theorem} △ A B E ≅ △ C D E

Since corresponding parts of congruent triangles are congruent, we know that the other sides of the triangles are congruent as well.

\underline{Definition of Congruent Triangles} \overline{A E} ≅ \overline{C E}; \overline{B E} ≅ \overline{D E}

Now, by the definition of a bisector, we can complete our proof.

\underline{Definition of Bisector} \overline{A C} and \overline{B D} bisect each other at E .

By combining these steps, let's complete the two-column proof.

Statement heading	Reason heading
$A B C D$ is a parallelogram.	Given
$\overline{A B} ∥ \overline{D C}$	a. Definition of Parallelogram
$∠ 1 ≅ ∠ 4;$ $∠ 2 ≅ ∠ 3$	b. Alternate Interior Angles Theorem
$\overline{A B} ≅ \overline{D C}$	c. Parallelogram Opposite Sides Theorem
d. $△ A B E ≅ △ C D E$	Angle-Side-Angle Congruence Theorem
$\overline{A E} ≅ \overline{C E};$ $\overline{B E} ≅ \overline{D E}$	e. Definition of Congruent Triangles
$\overline{A C}$ and $\overline{B D}$ bisects each other at $E .$	Definition of Bisector

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