Diagonals of a parallelogram bisect each other.


A parallelogram is a quadrilateral with opposite sides parallel and equal in length. It also has opposite angles equal in measure.

In a parallelogram, there are two pairs of opposite sides and two pairs of opposite angles. The two pairs of opposite sides are parallel, and the two pairs of opposite angles are equal.

The diagonals of a parallelogram are the line segments that connect opposite corners or vertices of the parallelogram. These diagonals intersect each other inside the parallelogram.

When the diagonals of a parallelogram bisect each other:
- The point of intersection of the diagonals divides each diagonal into two equal halves.
- This means that each diagonal is divided into two segments of equal length.
- The two line segments created by the intersection of the diagonals are congruent.

Proof that the diagonals of a parallelogram bisect each other:
To prove that the diagonals of a parallelogram bisect each other, we can use the concept of midpoint.

Let ABCD be a parallelogram, with diagonals AC and BD intersecting at point E.

Step 1:
Draw line segments AE, BE, CE, and DE.

Step 2:
Since ABCD is a parallelogram, AB is parallel to CD, and AD is parallel to BC.

Step 3:
By the properties of a parallelogram, AB is congruent to CD, and AD is congruent to BC.

Step 4:
Since AB is congruent to CD and AD is congruent to BC, we can conclude that triangle ABE is congruent to triangle CDE by side-side-side congruence.

Step 5:
By triangle congruence, angle AEB is congruent to angle CED.

Step 6:
Since angle AEB is congruent to angle CED, we can conclude that AE is congruent to CE and BE is congruent to DE by the properties of congruent triangles.

Step 7:
By the midpoint theorem, we can conclude that the diagonals AC and BD bisect each other at point E.

Therefore, the diagonals of a parallelogram bisect each other.