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Parallelogram Classification

Rectangles, rhombuses, and squares are specific parallelograms.

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Parallelogram Classification

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.

ABCD is a rectangle if and only if ABCD.

  • A quadrilateral is a rhombus if and only if it has four congruent sides.

ABCD is a rhombus if and only if AB¯¯¯¯¯¯¯¯BC¯¯¯¯¯¯¯¯CD¯¯¯¯¯¯¯¯AD¯¯¯¯¯¯¯¯.

  • 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.

ABCD is a square if and only if ABCD and AB¯¯¯¯¯¯¯¯BC¯¯¯¯¯¯¯¯CD¯¯¯¯¯¯¯¯AD¯¯¯¯¯¯¯¯.

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.

ABCD is parallelogram. If AC¯¯¯¯¯¯¯¯BD¯¯¯¯¯¯¯¯, then ABCD is also a rectangle.

2.  A parallelogram is a rhombus if the diagonals are perpendicular.

ABCD is a parallelogram. If AC¯¯¯¯¯¯¯¯BD¯¯¯¯¯¯¯¯, then ABCD is also a rhombus.

3.  A parallelogram is a rhombus if the diagonals bisect each angle.

ABCD is a parallelogram. If AC¯¯¯¯¯¯¯¯ bisects BAD and BCD and BD¯¯¯¯¯¯¯¯ bisects ABC and ADC, then ABCD is also a rhombus.

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 135, so the angles are not congruent. This is a rhombus.

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 SQRE.

A square has all the properties of a parallelogram, rectangle and rhombus.

Properties of a Parallelogram Properties of a Rhombus Properties of a Rectangle
  • SQ¯¯¯¯¯¯¯ER¯¯¯¯¯¯¯¯
  • SQ¯¯¯¯¯¯¯ER¯¯¯¯¯¯¯¯SE¯¯¯¯¯¯¯QR¯¯¯¯¯¯¯¯
  • mSER=mSQR=mQSE=mQRE=90
  • SE¯¯¯¯¯¯¯QR¯¯¯¯¯¯¯¯
  • SR¯¯¯¯¯¯¯QE¯¯¯¯¯¯¯¯
  • SEQQERSQEEQR
  • SR¯¯¯¯¯¯¯QE¯¯¯¯¯¯¯¯
  • QSRRSEQRSSRE
  • SA¯¯¯¯¯¯¯AR¯¯¯¯¯¯¯¯QA¯¯¯¯¯¯¯¯AE¯¯¯¯¯¯¯¯

All the bisected angles are 45.

Review

  1. RACE is a rectangle. Find:
    1. RG
    2. AE
    3. AC
    4. EC
    5. mRAC

  2. DIAM is a rhombus. Find:
    1. MA
    2. MI
    3. DA
    4. mDIA
    5. mMOA

  3. CUBE is a square. Find:
    1. mUCE
    2. mEYB
    3. mUBY
    4. mUEB

For questions 4-15, determine if the quadrilateral is a parallelogram, rectangle, rhombus, square or none.

For questions 16-19 determine if the following are ALWAYS, SOMETIME, or NEVER true. Explain your reasoning.

  1. A rectangle is a rhombus.
  2. A square is a parallelogram.
  3. A parallelogram is regular.
  4. A square is a rectangle.

Review (Answers)

To see the Review answers, open this PDF file and look for section 6.5. 

Vocabulary

rectangle

A parallelogram is a rectangle if and only if it has four right (congruent) angles: {{Inline image |source=Image:geo-0603-02b.png|size=125px}}

rhombus

A parallelogram is a rhombus if and only if it has four congruent sides: {{Inline image |source=Image:geo-0603-03b.png|size=125px}}

square

A parallelogram is a square if and only if it has four right angles and four congruent sides. {{Inline image |source=Image:geo-0603-04b.png|size=100px}}

converse

If a conditional statement is p \rightarrow q (if p, then q), then the converse is q \rightarrow p (if q, then p. Note that the converse of a statement is not true just because the original statement is true.

Parallelogram

A parallelogram is a quadrilateral with two pairs of parallel sides.

Reflexive Property of Congruence

\overline{AB} \cong \overline{AB} or \angle B \cong \angle B

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