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Parallelogram Classification

Rectangles, rhombuses, and squares are specific parallelograms.

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Parallelogram Classification

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? After completing this Concept, you'll be able to further classify a parallelogram based on its diagonals, angles, and sides.

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CK-12 Classifying Parallelograms

Guidance

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 \angle A \cong \angle B \cong \angle C \cong \angle D .

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

ABCD is a rhombus if and only if \overline{AB} \cong \overline{BC} \cong \overline{CD} \cong \overline{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 \angle A \cong \angle B \cong \angle C \cong \angle D and \overline{AB} \cong \overline{BC} \cong \overline{CD} \cong \overline{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 \overline{AC} \cong \overline{BD} , then ABCD is also a rectangle.

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

ABCD is a parallelogram. If \overline{AC} \perp \overline{BD} , then ABCD is also a rhombus.

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

ABCD is a parallelogram. If \overline{AC} bisects \angle BAD and \angle BCD and \overline{BD} bisects \angle ABC and \angle ADC , then ABCD is also a rhombus.

Example A

What typed of parallelogram are the figures below?

a)

b)

Answer:

a) All sides are congruent and one angle is 135^\circ , so the angles are not congruent. This is a rhombus.

b) All four angles are congruent but the sides are not. This is a rectangle.

Example B

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 C

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
  • \overline{SQ} \| \overline{ER}
  • \overline{SQ} \cong \overline{ER} \cong \overline{SE} \cong \overline{QR}
  • m \angle SER = m \angle SQR = m \angle QSE = m \angle QRE  = 90^\circ
  • \overline{SE} \| \overline{QR}
  • \overline{SR} \perp \overline{QE}
  •  \angle SEQ \cong \angle QER \cong \angle SQE \cong \angle EQR
  • \overline{SR} \cong \overline{QE}
  • \angle QSR \cong \angle RSE \cong \angle QRS \cong \angle SRE
  • \overline{SA} \cong \overline{AR} \cong \overline{QA} \cong \overline{AE}

All the bisected angles are 45^\circ .

CK-12 Classifying Parallelograms

Guided Practice

1. Is a rectangle SOMETIMES, ALWAYS, or NEVER a parallelogram? Explain why.

2. Is a rhombus SOMETIMES, ALWAYS, or NEVER equiangular? Explain why.

3. Is a quadrilateral SOMETIMES, ALWAYS, or NEVER a pentagon? Explain why.

Answers:

1. A rectangle has two sets of parallel sides, so it is ALWAYS a parallelogram.

2. Any quadrilateral, including a rhombus, is only equiangular if all its angles are 90^\circ . This means a rhombus is SOMETIMES equiangular, only when it is a square.

3. A quadrilateral has four sides, so it will NEVER be a pentagon with five sides.

Practice

  1. RACE is a rectangle. Find:
    1. RG
    2. AE
    3. AC
    4. EC
    5. m \angle RAC

  2. DIAM is a rhombus. Find:
    1. MA
    2. MI
    3. DA
    4. m \angle DIA
    5. m \angle MOA

  3. CUBE is a square. Find:
    1. m \angle UCE
    2. m \angle EYB
    3. m \angle UBY
    4. m \angle UEB

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.

Vocabulary

rectangle

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

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

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

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

Parallelogram

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

Reflexive Property of Congruence

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

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