# Properties of a Rhombus

How to solve the rhombus problems by using its properties: definition, properties of its diagonals and angles, examples, and their solutions.

## Definition

A rhombus is a parallelogram

whose sidea are all congruent.

So a rhombus

has all the properties of a parallelogram.

Properties of a parallelogram

## Property: Diagonals

For a rhombus,

the diagonals perpendicularly bisect each other.

## Example 1

The given figure is a rhombus.

So its sides are all congruent.*AD* = 5

So *BC* = 5.

The diagonals perpendicularly bisect each other.

So ∠*CPB* is a right angle.

See △*PBC*.

It's a right triangle.

And starting from the shortest side,

the sides are (3, *PB*, 5).

So △*PBC* is a (3, 4, 5) right triangle.

Pythagorean triples

So *PB* = 4.

*DP* = *PB* = 4

So *BD* = 4 + 4

= 8.

## Property: Angles

A rhombus is also a parallelogram.

So, for a rhombus,

its opposite interior angles are congruent.

Properties of a parallelogram - Angles

For a rhombus,

the diagonals bisect the interior angles.

## Example 2

The given figure is a rhombus.

∠*ADC* and ∠*ABC* are the opposite angles.

So these two angles are congruent.

Next, its diagonal *DB* bisects ∠*ADC* and ∠*ABC*.

So ∠[blue dot] are all congruent.

m∠[blue dot] = 50.

So m∠*ABC* = 2⋅m∠[blue dot]

= 2⋅[50]

= 100.

The other diagonal *AC* bisects ∠*DAB*.

For a rhombus,

the diagonals perpendicularly bisect each other.

So *AC* and *DB* perpendicularly bisect each other.

So ∠*DPA* is a right angle.

△*DPA* is a triangle.

m∠[blue dot] = 50

m∠*DPA* = 90

So m∠[green dot] + [50] + [90] = 180.

Interior angles of a triangle

50 + 90 = 140

Move +140 to the right side.

Then m∠[green dot] = 40.

m△*PAB* = m∠[green dot] = 40

So m△*PAB* = 40.