Geometry

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

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.

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.

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: