Properties of a Rhombus

Learn about the special parallelogram with 4 equal sides

What is a Rhombus?

Definition:

A rhombus is a parallelogram with 4 equal sides

Think of it as a square pushed over to one side, or a "diamond shape"

Look at rhombus EFGH below. Notice the tick marks showing ALL sides are equal:

Rhombus EFGH - All 4 sides are equal: EF = FG = GH = HE

Key Property:

$EF = FG = GH = HE$

(All 4 sides are equal - this is what makes it a rhombus!)

Rhombus vs Parallelogram

A rhombus is a special type of parallelogram. It has all the properties of a parallelogram, PLUS the extra property that all 4 sides are equal.

Parallelogram

Opposite sides equal

2 pairs of parallel sides
Opposite sides equal

Rhombus

ALL 4 sides equal

2 pairs of parallel sides
ALL 4 sides equal

Remember: Every rhombus is a parallelogram, but not every parallelogram is a rhombus!

All Properties of a Rhombus

A rhombus is a special parallelogram: it has all the properties of a parallelogram, and all four of its sides are equal:

Property 1: 4 Equal Sides

PQ = QR = RS = SP (all sides equal)

Property 2: 2 Pairs of Parallel Sides

PQ // SR and PS // QR

$PQ \parallel SR$ and $PS \parallel QR$

Property 3: Opposite Angles are Equal

∠T = ∠V = 45° (opposite) and ∠U = ∠W = 135° (opposite)

$\angle T = \angle V$ and $\angle U = \angle W$

Property 4: Adjacent Angles Sum to 180°

∠A + ∠B = 55° + 125° = 180°

$\angle A + \angle B = 180°$ (angles between two parallel sides)

Worked Examples

Example 1: Finding angles using opposite angles property

JKLM is a rhombus. ∠KLM = 135°. Find ∠e, ∠f, and ∠g.

Find angles e, f, and g

Solution:

Finding ∠e: Opposite angles are equal

$\angle e = \angle KLM = 135°$

Finding ∠f: Adjacent angles sum to 180°

$\angle f = 180° - 135° = 45°$

Finding ∠g: ∠g and ∠f are opposite angles

$\angle g = \angle f = 45°$

Answer: $\angle e = 135°$ , $\angle f = 45°$ , $\angle g = 45°$

Example 2: Finding angles in a diamond-shaped rhombus

EFGH is a rhombus. ∠HEF = 40°. Find ∠FGH, ∠EFG and ∠GHE.

Rhombus EFGH with ∠HEF = 40°: find the other three angles

Solution:

Step 1: Find ∠FGH (opposite to ∠HEF)

$\angle FGH = \angle HEF = 40°$

Step 2: Find ∠EFG (adjacent to ∠HEF)

$\angle EFG = 180° - 40° = 140°$

Step 3: Find ∠GHE (opposite to ∠EFG)

$\angle GHE = \angle EFG = 140°$

Step 4: Check

The four angles should add up to 360°:

$40° + 140° + 40° + 140° = 360°$ ✓

Answer: $\angle FGH = 40°$ , $\angle EFG = 140°$ , $\angle GHE = 140°$

Practice Problems

Practice 1: Find the unknown angles

ABCD is a rhombus. ∠ABC = 70°. Find ∠BCD and ∠DAB.

Find ∠BCD and ∠DAB

ShowHide solution

∠BCD (adjacent to ∠ABC):

$\angle BCD = 180° - 70° = 110°$

∠DAB (opposite to ∠BCD):

$\angle DAB = \angle BCD = 110°$

Answer: ∠BCD = 110°, ∠DAB = 110°

Practice 2: Using rhombus properties

PQRS is a rhombus. ∠SPQ = 55°. Find ∠x and ∠y.

PQRS is a rhombus with ∠SPQ = 55°

ShowHide solution

Finding ∠x (opposite to ∠SPQ):

$\angle x = \angle SPQ = 55°$

Finding ∠y (adjacent to ∠SPQ):

$\angle y = 180° - 55° = 125°$

Answer: $\angle x = 55°$ , $\angle y = 125°$

Practice 3: Diamond orientation

WXYZ is a rhombus. ∠WXY = 50°. Find all other angles.

Find ∠YZW, ∠ZWX, and ∠XYZ

ShowHide solution

∠ZWX (opposite to ∠XYZ):

First find ∠XYZ: Adjacent to ∠WXY, so $\angle XYZ = 180° - 50° = 130°$

$\angle ZWX = \angle XYZ = 130°$

∠YZW (opposite to ∠WXY):

$\angle YZW = \angle WXY = 50°$

Answer: ∠XYZ = 130°, ∠YZW = 50°, ∠ZWX = 130°

Practice 4: Identify properties

In rhombus KLMN, which side is equal to KL? Which angle is equal to ∠KLM?

Rhombus KLMN

ShowHide solution

Side equal to KL:

In a rhombus, ALL sides are equal!

$KL = LM = MN = NK$

Angle equal to ∠KLM:

∠MNK is opposite to ∠KLM

$\angle MNK = \angle KLM$

Answer: LM, MN and NK are all equal to KL. ∠MNK = ∠KLM

Key Takeaways

A rhombus has 4 equal sides
A rhombus is a special type of parallelogram
It has 2 pairs of parallel sides
Opposite angles are equal
Adjacent angles sum to 180°
All four angles add up to 360°