# 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

whose two pairs of opposite sides are parallel.

## Property: Sides

### Property

For a parallelogram,
the opposite sides are congruent.

## Example 1

### Solution

The given quadrilateral is a parallelogram.

So these two opposite sides
are congruent.

So [2x + 1] = .

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] = .

Move +4 to the right side.

Then y = 6 - 4 = 2.

x = 4
y = 2

Write these below.

So
x = 4
y = 2

## Example 2

### 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.

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

### Solution

The given quadrilateral is a parallelogram.

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

So m∠A +  = 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

## Property: Diagonals

### Property

For a parallelogram,
the diagonals bisect each other.

## Example 4

### 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] = .

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] = .

Divide both sides by 2.

Then y = 5.

x = 3
y = 5

Write these below.

So
x = 3
y = 5