Properties of Rhombuses, Rectangles, and Squares

Standard

G.CO.11 Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

Goals for this section:

Rhombuses, rectangles, and squares are all parallelograms that have special properties about them.

Rhombuses

It is a parallelogram with four congruent sides (equilateral).

  • Notice that the angles are not equiangular.

Diagonal properties of a Rhombus

Diagonals are perpendicular

If a parallelogram is a rhombus, then the diagonals form a right angle.

Diagonals bisect angles

If a parallelogram is a rhombus, then the diagonals bisect a pair of opposite angles.

  • Remember, bisect means to cut in half.

Rectangles

It is a parallelogram with four right angles (equiangular).

  • Notice it is not equilateral.

Diagonal Property of a Rectangle

The diagonals of a rectangle are congruent. So AC = BD.

Squares

It is a parallelogram with four right angles and four congruent sides.

Squares have the same properties as parallelograms, rhombuses, and rectangles!

Recall:

Remember that these are all parallelograms so they still have the properties that every parallelogram has! Here's a little summary of the properties:

Rhombuses

  • Equilateral
  • Diagonals are perpendicular

Additional parallelogram properties:

  • Opposite sides are parallel
  • Opposite angles are congruent
  • Diagonals bisect each other

Rectangles

  • Equiangular
  • Diagonals are congruent

Additional parallelogram properties:

  • Opposite sides are parallel
  • Opposite sides are congruent
  • Diagonals bisect each other

Squares

  • Equilateral and equiangular
  • Diagonals are perpendicular and congruent

Additional parallelogram properties:

  • Opposite sides are parallel
  • Diagonals bisect each other