Rhombus: Property

How to use the properties of a rhombus to solve the related problems: definition, 2 properties (diagonals, angles), 2 examples, and their solutions.

Definition

Definition

A rhombus is a parallelogram
whose sides are all congruent.

So a rhombus
has all the properties of a parallelogram.

Property: Diagonals

Property

For a rhombus,
the diagonals perpendicularly bisect each other.

Example 1

Example

Solution

If ABCD is a rhombus,
then the sides are congruent.

AD = 5

So BC = 5.

If ABCD is a rhombus,
then the diagonals
perpendicularly bisect each other.

See △PBC.
It's a right triangle.

The sides are (3, PB, 5).
So this right triangle is
a (3, 4, 5) right triangle.

Pythagorean Triple

So PB = 4.

PB = 4
DP = PB

So DP = 4.

DP = 4
PB = 4

So BD = 4 + 4 = 8.

BC = 5
BD = 8

Write these below.

So
BC = 5
BD = 8
is the answer.

Property: Angles

Property

For a rhombus,
the opposite interior angles are congruent.

(This is also true for a parallelogram.)

And the diagonals bisect the interior angles.

Example 2

Example

Solution

The given quadrilateral is a rhombus.
DB is a diagonal.

So DB bisects ∠ADC.

∠ADP is 50º.

Then ∠PDC is also 50º.

∠ADC is, 50 + 50, 100º.

∠ADC and ∠ABC are the opposite angles.

So ∠ABC is also, 50 + 50, 100º.

So m∠ABC = 100.

Next, AC and DB are the diagonals.
They are perpendicular.
So draw a right angle like this.

See △ADP.

The interior angles are
∠DAP, 50º, and 90º.

So m∠DAP + [50] + [90] = 180.

Triangle: Interior Angles

+50 + 90 = +140

Move +140 to the right side.

Then m∠DAP = 40.

Write 40º on ∠DAP.

AC is a diagonal
that bisects ∠DAB.

∠DAP is 40º.

So ∠PAB is also 40º.

So m∠PAB = 40.

Write m∠ABC and m∠PAB below.

So
m∠ABC = 100
m∠PAB = 40
is the answer.