Parallelogram

A quadrilateral is denoted as . In a quadrilateral:
  • Opposite sides are referred to as opposite sides.
  • Opposite angles are referred to as opposite angles.

Properties of a Parallelogram

In a parallelogram:
  1. The lengths of two pairs of opposite sides are equal.
  2. The measures of two pairs of opposite angles are equal.
  3. The two diagonals bisect each other.

[Explanation]

1. Opposite sides and angles of a parallelogram are equal in length and measure.

In parallelogram , draw the diagonal .
Consider and :
  • The diagonal is common to both triangles.
  • Since and :
By the ASA congruence condition, .
Thus:
Additionally, from the above congruence:
Hence, in a parallelogram, both pairs of opposite sides are equal in length, and both pairs of opposite angles are equal in measure.

2. The diagonals of a parallelogram bisect each other.

Let the diagonals of parallelogram intersect at point .
Consider and :
  • Since :
By the ASA congruence condition, .
Thus:
Therefore, the diagonals of a parallelogram bisect each other.