Properties of Parallelograms

A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel. Rectangles, rhombuses, and squares are all special types of parallelograms — but the basic parallelogram doesn't need equal sides or right angles.

Angle Properties

The four angles of any quadrilateral add up to 360°. In a parallelogram, the angles have additional structure:

Opposite angles are equal. In parallelogram ABCD:

$$\angle A = \angle C \quad \text{and} \quad \angle B = \angle D$$

Consecutive angles are supplementary. Any two angles that share a side add up to 180°:

$$\angle A + \angle B = 180° \quad \angle B + \angle C = 180°$$ $$\angle C + \angle D = 180° \quad \angle A + \angle D = 180°$$

This makes sense because the parallel sides act like parallel lines cut by a transversal, and co-interior angles always sum to 180°.

Side Properties

Opposite sides are parallel (that's the definition):

$$\text{AD} \parallel \text{BC} \quad \text{and} \quad \text{AB} \parallel \text{CD}$$

Opposite sides are also equal in length:

$$\text{AD} = \text{BC} \quad \text{and} \quad \text{AB} = \text{CD}$$

Diagonal Properties

The two diagonals of a parallelogram bisect each other — they cross at their mutual midpoint. If E is the intersection point:

$$AE = EC \quad \text{and} \quad DE = EB$$

Drawing one diagonal also splits the parallelogram into two congruent triangles. The diagonal acts as a transversal, so the alternate interior angles it creates are equal — that's what makes the two triangles congruent by ASA.

Example

In parallelogram WXYZ, ∠X = \(4a - 40\) and ∠Z = \(2a - 8\). Find ∠W.

Since X and Z are opposite angles, they must be equal:

$$4a - 40 = 2a - 8$$ $$2a = 32$$ $$a = 16$$

So ∠X = \(4(16) - 40 = 24°\).

Since W and X are consecutive angles, they're supplementary:

$$\angle W = 180 - 24 = 156°$$

You can verify: ∠W = ∠Y = 156° and ∠X = ∠Z = 24°. The sum is \(156 + 24 + 156 + 24 = 360°\). ✓