# 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.