Lesson 2: Parallelograms

Property

A parallelogram is a quadrilateral with two pairs of opposite sides that are parallel, which gives it the following properties:

• Opposite sides are congruent (equal in length).
• Opposite angles are congruent (equal in measure).
• Consecutive angles are supplementary (their sum is $180^\circ$).
• The diagonals bisect each other (they cut each other into two equal halves).

Examples

Property

The area of a parallelogram can be determined without the formula using two visual methods:

1. Decomposition: Decompose the parallelogram into a triangle and a trapezoid. Rearrange these pieces to form a rectangle. The area is the length times the width of this new rectangle.
2. Enclosure: Enclose the parallelogram within a larger rectangle. The area of the parallelogram is the area of the enclosing rectangle minus the areas of the two congruent right triangles formed at either end.

Examples

• Decomposition: A parallelogram has a base of $8$ units and a height of $5$ units. By cutting a right triangle from one side and moving it to the other, we form a rectangle with dimensions $8 \times 5$. The area is $8 \times 5 = 40$ square units.
• Enclosure: A parallelogram with a base of $10$ units and a height of $6$ units is enclosed in a rectangle. The rectangle has a width of $10+4=14$ units and a height of $6$ units. The area of the parallelogram is the area of the rectangle minus the area of the two triangles: $(14 \times 6) - 2 \times (\frac{1}{2} \times 4 \times 6) = 84 - 24 = 60$ square units.

Explanation

These methods demonstrate why the area formula for a parallelogram works. The decomposition method transforms the parallelogram into a rectangle with the same base, height, and area. The enclosure method calculates the area indirectly by subtracting the excess area from a larger, simpler shape. Both techniques are powerful visual strategies for understanding the concept of area conservation.

Property

A parallelogram can be rearranged into a rectangle with the same base and height. Choose any side of the parallelogram as the base (length $b$), and let $h$ be the perpendicular distance between the base and the opposite side.
The area is given by the formula:

Examples

• A parallelogram has a base of 12 cm and a height of 5 cm. Its area is $12 \times 5 = 60$ square cm.
• A section of a patio is shaped like a parallelogram with a base of 8 feet and a height of 6 feet. The area is $8 \times 6 = 48$ square feet.
• Even if the slanted side is 9 inches, if the base is 15 inches and the height is 7 inches, the area is $15 \times 7 = 105$ square inches.

Explanation

Think of a parallelogram as a slanted rectangle. By slicing off a triangle from one end and moving it to the other, you create a perfect rectangle. This new rectangle has the same base and height, which is why the area formula works!