Properties of a Parallelogram

Learn the special properties of parallelograms - sides, angles, and parallel lines

What is a Parallelogram?

Definition:

A parallelogram is a quadrilateral with 2 pairs of parallel sides

Look at parallelogram WXYZ below. Notice the arrow markers showing which sides are parallel:

Parallelogram WXYZ with parallel sides marked

Parallel Sides:

$WX \parallel ZY$ (top // bottom)
$WZ \parallel XY$ (left // right)

Equal Sides:

$WX = ZY$ (opposite sides)
$WZ = XY$ (opposite sides)

Angle Properties of a Parallelogram

Property 1: Opposite Angles are Equal

In a parallelogram, angles that are diagonally opposite each other are equal.

Opposite angles are equal: ∠A = ∠C = 120° and ∠B = ∠D = 60°

$\angle A = \angle C$ (opposite angles)

$\angle B = \angle D$ (opposite angles)

Property 2: Adjacent Angles Sum to 180°

Any two angles that are next to each other (adjacent) add up to 180°. This is because each such pair lies between two parallel sides.

Adjacent angles sum to 180°: ∠E + ∠F = 115° + 65° = 180°

All adjacent pairs sum to 180°:

$\angle E + \angle F = 180°$

$\angle F + \angle G = 180°$

$\angle G + \angle H = 180°$

$\angle H + \angle E = 180°$

Why Do Adjacent Angles Sum to 180°?

The top and bottom sides are parallel, and the left side runs from one to the other. The two angles at its ends lie between the parallel sides, and two such angles always add up to 180°.

∠P and ∠S lie between the parallel sides: 130° + 50° = 180°

Identifying Parallel and Equal Sides

In a parallelogram, you need to be able to identify:

Which sides are parallel to each other
Which sides are equal in length
Which angles are opposite (and therefore equal)
Which angles are adjacent (and therefore sum to 180°)

Parallelogram JKLM - Can you identify the relationships?

Parallel Sides:

$JK \parallel ML$

$JM \parallel KL$

Opposite Angles:

$\angle j = \angle l$

$\angle k = \angle m$

Worked Examples

Example 1: Finding Opposite Angles

EFGH is a parallelogram. ∠HEF = 120°. Find ∠j, ∠k, and ∠m.

Find angles j, k, and m

Solution:

Finding ∠k: Opposite angles are equal

$\angle k = \angle HEF = 120°$

Finding ∠m: Adjacent angles sum to 180°

$\angle m = 180° - 120° = 60°$

Finding ∠j: ∠j and ∠m are opposite angles

$\angle j = \angle m = 60°$

Answer: $\angle j = 60°$ , $\angle k = 120°$ , $\angle m = 60°$

Example 2: Using Adjacent Angles

PQRS is a parallelogram. ∠PSR = 51°. Find ∠a.

PQRS is a parallelogram with ∠PSR = 51°

Solution:

∠a (that is, ∠SPQ) and ∠PSR are adjacent angles: they lie between the parallel sides PQ and SR.

Adjacent angles of a parallelogram add up to 180°, so

$\angle a = 180° - 51° = 129°$

Answer: $\angle a = 129°$

Practice Problems

Practice 1: Find the unknown angles

ABCD is a parallelogram. ∠ABC = 75°. Find ∠BCD, ∠CDA, and ∠DAB.

Find all unknown angles

ShowHide solution

∠CDA (opposite to ∠ABC):

$\angle CDA = \angle ABC = 75°$ (opposite angles)

∠BCD (adjacent to ∠ABC):

$\angle BCD = 180° - 75° = 105°$

∠DAB (opposite to ∠BCD):

$\angle DAB = \angle BCD = 105°$

Answer: ∠BCD = 105°, ∠CDA = 75°, ∠DAB = 105°

Practice 2: Identify parallel sides

In parallelogram MNOP, which side is parallel to MN? Which side is equal to NO?

Parallelogram MNOP

ShowHide solution

Side parallel to MN: PO (opposite side)

$MN \parallel PO$

Side equal to NO: MP (opposite side)

$NO = MP$

Answer: MN // PO and NO = MP

Practice 3: Finding angles using properties

WXYZ is a parallelogram. ∠WXY = 118°. Find ∠XYZ.

Find ∠XYZ

ShowHide solution

∠WXY and ∠XYZ are adjacent angles

Adjacent angles in a parallelogram sum to 180°

$\angle XYZ = 180° - 118°$

$\angle XYZ = 62°$

Key Takeaways

A parallelogram has 2 pairs of parallel sides
Opposite sides are equal in length
Opposite angles are equal
Adjacent angles sum to 180° (they lie between two parallel sides)
All four angles add up to 360°