Parallelogram: Property

How to use the properties of a parallelogram to solve the related problems: definition, 3 properties (sides, angles, diagonals), 4 examples, and their solutions.

Definition

Definition

A parallelogram is a quadrilateral
whose two pairs of opposite sides are parallel.

Property: Sides

Property

For a parallelogram,
the opposite sides are congruent.

Example 1

Example

Solution

The given quadrilateral is a parallelogram.

So these two opposite sides
are congruent.

So [2x + 1] = [9].

Move +1 to the other side.

Then 2x = 9 - 1 = 8.

Divide both sides by 2.

Then x = 4.

The other two opposite sides
are also congruent.

So [y + 4] = [6].

Move +4 to the right side.

Then y = 6 - 4 = 2.

x = 4
y = 2

Write these below.

So
x = 4
y = 2
is the answer.

Example 2

Example

Solution

If OPQR is a parallelogram,
the opposite sides are parallel and congruent.

Then see PQ and OR.

You can also choose OP and RQ.
You'll get the same answer.

See OR.

O(0, 0)
R(6, 2)

The change of x is +6.
The change of y is +2.

Then PQ will also show these changes.

From point P,
the change of x is +6
and the change of y is +2.

This is true because
OR and PQ are parallel and congruent.

P(1, 4)

So Q(1 + 6, 4 + 2).

Translation: Point

1 + 6 = 7
4 + 2 = 6

So Q(7, 6).

Property: Angles

Property

For a parallelogram,
the opposite interior angles are congruent.

m∠1 = m∠1'
m∠2 = m∠2'

And the adjacent pair of angles are supplementary.

m∠[blue] + m∠[green] = 180

Example 3

Example

Solution

The given quadrilateral is a parallelogram.

∠A and ∠D are a pair of adjacent angles.
m∠D = 70

So m∠A + [70] = 180.

Move +70 to the right side.

Then m∠A = 110.

Write 110º on ∠A.

∠B and ∠D are the opposite angles.
∠D is 70º

So ∠B is 70º.

∠A and ∠C are the opposite angles.
∠A is 110º

So ∠C is 110º.

Write m∠A, m∠B, and m∠C.

So
m∠A = 110
m∠B = 70
m∠C = 110
is the answer.

Property: Diagonals

Property

For a parallelogram,
the diagonals bisect each other.

Example 4

Example

Solution

The given quadrilateral is a parallelogram.

So the diagonals bisect each other.

The blue diagonal is bisected
by the other diagonal.

So [3x - 2] = [7].

Move -2 to the right side.

Then 3x = 9.

Divide both sides by 3.

Then x = 3.

Next, the brown diagonal is also bisected
by the other diagonal.

So [2y] = [10].

Divide both sides by 2.

Then y = 5.

x = 3
y = 5

Write these below.

So
x = 3
y = 5
is the answer.