Properties of a Parallelogram

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.

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.

For a parallelogram,
the parallel sides are congruent.

In other words,
the opposite sides are congruent.

Example 1

Find the values of x and y.

The given figure is a parallelogram.

So These two opposite sides are congruent.

So 2x + 1 = 9.

Move +1 to the right side.

Then 2x = 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

If OPQR is a parallelogram, find the coordinates of point Q. P(1, 4), R(6, 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

If OPQR is a parallelogram, find the coordinates of point Q. P(1, 4), R(6, 2).

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

The opposite interior angles are congruent. The adjacent interior angles are supplementary.

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

Find the measures of angle A, B, and C. [measure of angle D] = 70

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.

For a parallelogram,
the diagonals bisect each other.

Example 4

Find the values of x and y.

The given figure is a parallelogram.
So their diagonals bisect each other.

First, see the blue segments.
Then [3x - 2] = [7].

Move -2 to the right side.

Then 3x = 9.

Divide both sides by 3.

Then x = 3.

The other diagonal is bisected.

So [2y] = [10].

Didide both sides by 2.

Then y = 5.

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