A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel.

If you did the previous reading in this unit, you'll recognize that this definition is similar to the definition of a trapezoid. The difference is in the number of pairs of sides which are parallel.

Here's a picture of a parallelogram:
 

Parallelograms have some very interesting properties, and we'll talk about several of them here.

Congruent Sides: In a parallelogram, not only are the opposite sides parallel, they're also congruent.

Supplementary Angles: In a parallelogram, every pair of adjacent angles is supplementary.

Congruent Angles: If you think about the previous property for a minute, you will hopefully realize that this means that in a parallelogram, the opposite angles are always congruent.

Congruent Triangles: A diagonal of a parallelogram divides the parallelogram into two congruent triangles. You can prove this using the Alternate Interior Angle theorem and Angle-Side-Angle.

More Congruent Triangles: If you construct both diagonals of a parallelogram, they split the parallelogram into four small triangles. The opposite pairs are congruent. (Alternate Interior Angle theorem and Side-Angle-Side will help you prove this).

Bisected Diagonals: A consequence of the previous property is that each diagonal bisects (cuts in half) the other one.

Area: The area of a parallelogram is simply the base times the height. Be careful, though; you can take any side as the base, but that doesn't mean the adjacent side is the height. The height is the length of a perpendicular segment between the base and the opposite side.

1.

Are any trapezoids parallelograms?

2.

Which of the following is true: "All parallelograms are quadrilaterals," or "All quadrilaterals are parallelograms."

3.

If one angle of a parallelogram is 30 degrees, what are the other angles?

4.

If one angle of a parallelogram is x, and another angle is 3x, what is the value of x?

5.

The opposite angles of a parallelogram are 2x + 10 and 4x - 20. What are the angles of the parallelogram?

6.

The adjacent angles of a parallelogram are in the ratio 3:5. What is the smaller angle?

7.

The sides of a parallelogram are 10 and 15. One of the angles of the parallelogram has measure 30 degrees. What is the area of the parallelogram?