# Properties of a Parallelogram

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

## Definition

A parallelogram is a quadrilateral

whose two pairs of opposite sides are parallel.

Parallel lines

## Property: Sides

For a parallelogram,

the parallel sides are congruent.

In other words,

the opposite sides are congruent.

## Example 1

The given figure is a parallelogram.

So These two opposite sides are congruent.

So 2*x* + 1 = 9.

Move +1 to the right side.

Then 2*x* = 8.

Divide both sides by 2.

Then *x* = 4.

These two opposite sides are also congruent.

So *y* + 4 = 6.

Move +4 to the right side.

Then *y* = 2.

So *x* = 4, *y* = 2 is the answer.

## Example 2

*OPQR* is a parallelogram.

So the opposite sides are parallel and congruent.

*OR* and *PQ* are the opposite sides.

So these two sides are parallel and congruent.

The change of the coordinates of *OR*

is (+6, +2).

So the change of the coordinates of *PQ*

is also (+6, +2).*PQ* starts from *P*(1, 4).

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

Translation of a point

By using the translation,

you can solve this problem easily,

rather than

using the slope formula and the distance formula.

Slope of a line

Distance formula

1 + 6 = 7

4 + 2 = 6

So *Q*(7, 6) is the answer.

## Example 2: Another Solution

You can also choose the other pair of opposite sides.*OPQR* is a parallelogram.

So the opposite sides are parallel and congruent.

*OP* and *RQ* are the opposite sides.

So these two sides are parallel and congruent.

The change of the coordinates of *OP*

is (+1, +4).

So the change of the coordinates of *RQ*

is also (+1, +4).*RQ* starts from *R*(6, 2).

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

6 + 1 = 7

2 + 4 = 6

So *Q*(7, 6) is the answer.

As you can see,

you can get the same answer.

## Property: Angles

There are two properties

of the angles of a parallelogram.

1. The opposite interior angles are congruent.

2. The adjacent interior angles are supplementary:

m∠[blue] + m∠[green] = 180

Supplementary angles

This is true because

the adjacent angles are

the consecutive interior angles in parallel lines.

Consecutive interior angles in parallel lines

## Example 3

∠*A* and ∠*D* are the adjacent angles.

m∠*D* = 70

So m∠*A* + 70 = 180.

Move +70 to the right side.

Then m∠*A* = 110.

∠*B* and ∠*D* are the opposite angles.

m∠*D* = 70

So m∠*B* = 70.

∠*A* and ∠*C* are the opposite angles.

m∠*A* = 110

So m∠*C* = 110.

So m∠*A* = 110, m∠*B* = 70, and m∠*C* = 110.

## Property: Diagonals

For a parallelogram,

the diagonals bisect each other.

## Example 4

The given figure is a parallelogram.

So their diagonals bisect each other.

First, see the blue segments.

Then [3*x* - 2] = [7].

Move -2 to the right side.

Then 3*x* = 9.

Divide both sides by 3.

Then *x* = 3.

The other diagonal is bisected.

So [2*y*] = [10].

Didide both sides by 2.

Then *y* = 5.

So *x* = 3, *y* = 5 is the answer.