The Parallelogram Diagonals Theorem: Proof and Explanation

Parallelogram Diagonals Theorem

The Parallelogram Diagonals Theorem states that the diagonals of a parallelogram bisect each other

The Parallelogram Diagonals Theorem states that the diagonals of a parallelogram bisect each other. This means that the point where the diagonals intersect divides each diagonal into two equal halves.

To better understand this theorem, let’s consider a parallelogram ABCD.

In this parallelogram, let’s denote the point where the diagonals intersect as point E.

We want to prove that segment AE is congruent to segment CE and segment BE is congruent to segment DE.

Proof:

Step 1: Draw the diagonals AD and BC of parallelogram ABCD.

Step 2: Connect points A and C with the point where the diagonals intersect, point E.

Step 3: In parallelogram ABCD, opposite sides are parallel. Therefore, segment AD is parallel to segment BC and segment AB is parallel to segment DC.

Step 4: Using the Alternate Interior Angles Theorem, we can conclude that angle AED and angle CEB are congruent, as they are alternate interior angles formed by parallel lines AD and BC and transversal AE.

Step 5: Similarly, using the Alternate Interior Angles Theorem, we can conclude that angle DEA and angle BEC are congruent.

Step 6: As angles AED and CEB are congruent, and angles DEA and BEC are congruent, we can apply the Angle-Angle (AA) Congruence Theorem to prove that triangle AED and triangle CEB are congruent.

Step 7: If triangle AED and triangle CEB are congruent, then segment AE is congruent to segment CE and segment DE is congruent to segment BE, according to the Side-Side-Side (SSS) Congruence Theorem.

Therefore, we have proved that the diagonals of a parallelogram bisect each other, i.e., segment AE is congruent to segment CE and segment DE is congruent to segment BE. This is known as the Parallelogram Diagonals Theorem.

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