Parallelograms

Parallelograms

A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel. Special relationships are a reality between the measures of consecutive angles, opposite angles and opposite sides of a parallelogram. The square is the most obvious parallelogram, because it has 2 sets of parallel sides.

Facts About a Parallelogram

(1) The degree measure of the four angles of a parallelogram add up to 360 degrees. Remember that all quadrilaterals (4 sided figures) have angles which add up to 360 degrees. Here's a sample:

Parallelogram Angles

Then: a + b + c + d = 360 degrees

(2) The degree measure of any two consecutive angles add up to 180 degrees. In parallelogram ABCD:

Consecutive angles in a parallelogram

angle a + angle b = 180 degrees
angle b + angle c = 180 degrees
angle c + angle d = 180 degrees
angle a + angle d = 180 degrees

(3) Opposite angles have the same measure in terms of degrees.

In parallelogram ABCD:

Opposite angles in a parallelogram

angle a = angle c
angle b = angle d

Sides of a Parallelogram

In parallelogram ABCD:

(1) Opposite sides are parallel:

Opposite sides in a parallelogram are parallel

side AD || side BC
side AB || side CD

NOTE: The symbol || means parallel.

(2) Opposite sides have the same lengths:

Opposite sides have the same lengths

side AD = side BC
side AB = side CD

Diagonals of a Parallelogram

The diagonals of a parallelogram divide the parallelogram into two side-by-side triangles. As shown in the picture below, diagonal AC forms equal alternate interior angles with each pair of parallel sides. We can also see that there are two triangles in the picture below. Triangle 1 is congruent to triangle 2 by ASA (Angle-Side-Angle) Method.

Where did the two triangles come from? They were formed by diagonal AC.

Cutting a parallelogram into two triangles

I should also note that diagonals of a parallelogram bisect each other as shown in the picture below.

Bisection of parallelogram diagonals

AE = EC
DE = EB
where E is the midpoint of BOTH diagonals.

Sample:

In parallelogram WXYZ, the measure of angle X = 4a - 40 and the measure of angle Z = 2a - 8. Find the measure of angle W?

Solution:

(1) Find the value of a.

Since angles X and Z are opposite angles of parallelogram WXYZ, they have the same measure. We equate the terms and solve for a.

4a - 40 = 2a - 8
4a - 2a = 40 - 8
2a = 32
a = 32/2
a = 16

(2) Since a = 16, we now find the measure of angle X by substituting 16 for a in 4a - 40.

angle X = 4a - 40
angle X = 4(16) - 40
angle X = 64 - 40
angle X = 24

(3) We also know that consecutive angles of a parallelogram are supplementary (add up to 180 degrees) and so, this fact is used to find the measure of angle W.

angle W = 180 - angle X
angle W = 180 - 24
angle W = 156

By Mr. Feliz
©2007

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