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.

**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

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 \(\overline{BE} \cong \overline{DE}\).

Parallelograms cont.

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.

- If opposite sides are congruent, then the quadrilateral is a parallelogram.
- If opposite angles are congruent, then the quadrilateral is a parallelogram.
- 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. - 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.

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.

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.

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.

Here is one possible way to identify special parallelograms: