# Parallelograms

## Big Picture

There are two main types of quadrilaterals: parallelograms and non-parallelograms. Parallelograms can also be divided into many sub-groups, including rectangles, trapezoids, and squares. Each type of parallelogram has its own unique set of properties.

## Key Terms

Quadrilateral: A polygon with four sides.

Parallelogram: A quadrilateral with two pairs of parallel sides.

Rectangle: Parallelogram with four right angles

Rhombus: Parallelogram with four congruent sides

Square: Rectangle with four congruent sides

## Properties of Parallelograms

Since parallelograms are quadrilaterals, the sum of the interior angles is 360°. Some examples of parallelograms include:

There are several theorems that describe the properties of parallelograms.Theorems:

• If a quadrilateral is a parallelogram, then opposite sides are congruent.
• If a quadrilateral is a parallelogram, then opposite angles are congruent.
• If a quadrilateral is a parallelogram, then consecutive angles are supplementary.

If ABCD is a parallelogram, then x° + y° = 180°.

4. If a quadrilateral is a parallelogram, then the diagonals bisect each other.

If ABCD is a parallelogram, then $$\overline{AE} \cong \overline{CE}$$ and $$\overline{BE} \cong \overline{DE}$$.

## Geometry

Parallelograms cont.

## Identifying Parallelograms

We can prove a quadrilaterial is a parallelogram by using the definition of a parallelogram.

• Definition: If both pairs of opposite sides are parallel, the quadrilateral is a parallelogram.

The converses of the theorems described above can be used to prove that a quadrilateral is a parallelogram.

1. If opposite sides are congruent, then the quadrilateral is a parallelogram.
2. If opposite angles are congruent, then the quadrilateral is a parallelogram.
3. If consecutive angles are supplementary, then the quadrilateral is a parallelogram.
a. This converse can be hard to apply - we would need to show that each angle is supplementary to its neighbor.
4. If diagonals bisect each other, then the quadrilateral is a parallelogram.

There is an additional theorem that shows a quadrilateral is a parallelogram:

• Theorem: If one set of parallel lines are congruent, then the quadrilateral is a parallelogram.

### Parallelogram in the Coordinate Plane

There are several ways to use the coordinate plane to show that a quadrilateral is a parallelogram:

Method 1: Use the distance formula to find the lengths of the sides and see if opposite sides are congruent.

Method 2: Find the slopes of the sides of the parallelogram.

• If the slopes of two opposite sides are equal, then they’re parallel.
• If both pairs of opposite sides are parallel, the figure is a parallelogram.

Method 3: Use the midpoint formula for each diagonal. If the midpoint is the same for both, the figure is a parallelogram.

Method 4: Determine if one pair of opposite sides has the same slope and the same length. If both statements are true, then the figure is a parallelogram.

## Special Parallelograms

There are three special types of parallelograms: rhombus, rectangle, and square. These special parallelograms can be identified with these theorems:

• Rhombus Theorem: A quadrilateral is a rhombus if and only if it has four congruent sides.
• Rectangle Theorem: A quadrilateral is a rectangle if and only if it has four right angles.
• Square Theorem: A quadrilateral is a square if and only if it has four right angles and four congruent sides.
• A square is ALWAYS a rectangle and a rhombus.

### Diagonals in Special Parallelograms

The diagonals of special parallelograms have additional properties that can be used to identify them:

• Theorem: A parallelogram is a rhombus if and only if the diagonals are perpendicular.
• Theorem: A parallelogram is a rhombus if and only if the diagonals bisect each angle.
• Theorem: A parallelogram is a rectangle if and only if the diagonals are congruent.

Since a square is a rectangle and a rhombus, it has the properties of a rhombus, rectangle, and parallelogram.

### Identifying Special Parallelograms in the Coordinate Plane

Here is one possible way to identify special parallelograms: