Classifying Parallelograms
Rectangles, rhombuses (also called rhombi) and squares are all more specific versions of parallelograms, also called special parallelograms.
 A quadrilateral is a rectangle if and only if it has four right (congruent) angles.
 A quadrilateral is a rhombus if and only if it has four congruent sides.
 A quadrilateral is a square if and only if it has four right angles and four congruent sides. By definition, a square is a rectangle and a rhombus.
You can always show that a parallelogram is a rectangle, rhombus, or square by using the definitions of these shapes. There are some additional ways to prove parallelograms are rectangles and rhombuses, shown below:
1. A parallelogram is a rectangle if the diagonals are congruent.
2. A parallelogram is a rhombus if the diagonals are perpendicular.
3. A parallelogram is a rhombus if the diagonals bisect each angle.
What if you were given a parallelogram and information about its diagonals? How could you use that information to classify the parallelogram as a rectangle, rhombus, and/or square?
Examples
Example 1
Is a rectangle SOMETIMES, ALWAYS, or NEVER a parallelogram? Explain why.
A rectangle has two sets of parallel sides, so it is ALWAYS a parallelogram.
Example 2
Is a quadrilateral SOMETIMES, ALWAYS, or NEVER a pentagon? Explain why.
A quadrilateral has four sides, so it will NEVER be a pentagon with five sides.
Example 3
What typed of parallelogram are the figures below?
For the first figure, all sides are congruent and one angle is
For the second figure, all four angles are congruent but the sides are not. This is a rectangle.
Example 4
Is a rhombus SOMETIMES, ALWAYS, or NEVER a square? Explain why.
A rhombus has four congruent sides and a square has four congruent sides and angles. Therefore, a rhombus is a square when it has congruent angles. This means a rhombus is SOMETIMES a square.
Example 5
List everything you know about the square
A square has all the properties of a parallelogram, rectangle and rhombus.
Properties of a Parallelogram  Properties of a Rhombus  Properties of a Rectangle 












All the bisected angles are
Review
RACE is a rectangle. Find:RG AE AC EC m∠RAC
DIAM is a rhombus. Find:MA MI DA m∠DIA m∠MOA
CUBE is a square. Find:m∠UCE m∠EYB m∠UBY m∠UEB
For questions 415, determine if the quadrilateral is a parallelogram, rectangle, rhombus, square or none.
For questions 1619 determine if the following are ALWAYS, SOMETIME, or NEVER true. Explain your reasoning.
 A rectangle is a rhombus.
 A square is a parallelogram.
 A parallelogram is regular.
 A square is a rectangle.
Review (Answers)
To see the Review answers, open this PDF file and look for section 6.5.