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:

Properties of Parallelograms

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.
opposite angles are congruent

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

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

 opposite sides are congruent

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.
Identifying Parallelograms

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

Identifying Special Parallelograms in the Coordinate Plane