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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.

\begin{align*}ABCD\end{align*} is a rectangle if and only if \begin{align*}\angle A \cong \angle B \cong \angle C \cong \angle D\end{align*}.

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

\begin{align*}ABCD\end{align*} is a rhombus if and only if \begin{align*}\overline{AB} \cong \overline{BC} \cong \overline{CD} \cong \overline{AD}\end{align*}.

  • 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.

\begin{align*}ABCD\end{align*} is a square if and only if \begin{align*}\angle A \cong \angle B \cong \angle C \cong \angle D\end{align*} and \begin{align*}\overline{AB} \cong \overline{BC} \cong \overline{CD} \cong \overline{AD}\end{align*}.

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.

\begin{align*}ABCD\end{align*} is parallelogram. If \begin{align*}\overline{AC} \cong \overline{BD}\end{align*}, then \begin{align*}ABCD\end{align*} is also a rectangle.

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

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

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

\begin{align*}ABCD\end{align*} is a parallelogram. If \begin{align*}\overline{AC}\end{align*} bisects \begin{align*}\angle BAD\end{align*} and \begin{align*}\angle BCD\end{align*} and \begin{align*}\overline{BD}\end{align*} bisects \begin{align*}\angle ABC\end{align*} and \begin{align*}\angle ADC\end{align*}, then \begin{align*}ABCD\end{align*} 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 \begin{align*}135^\circ\end{align*}, 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 \begin{align*}SQRE\end{align*}.

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

Properties of a Parallelogram Properties of a Rhombus Properties of a Rectangle
  • \begin{align*}\overline{SQ} \| \overline{ER}\end{align*}
  • \begin{align*}\overline{SQ} \cong \overline{ER} \cong \overline{SE} \cong \overline{QR}\end{align*}
  • \begin{align*}m \angle SER = m \angle SQR = m \angle QSE = m \angle QRE = 90^\circ\end{align*}
  • \begin{align*}\overline{SE} \| \overline{QR}\end{align*}
  • \begin{align*}\overline{SR} \perp \overline{QE}\end{align*}
  • \begin{align*} \angle SEQ \cong \angle QER \cong \angle SQE \cong \angle EQR\end{align*}
  • \begin{align*}\overline{SR} \cong \overline{QE}\end{align*}
  • \begin{align*}\angle QSR \cong \angle RSE \cong \angle QRS \cong \angle SRE\end{align*}
  • \begin{align*}\overline{SA} \cong \overline{AR} \cong \overline{QA} \cong \overline{AE}\end{align*}

All the bisected angles are \begin{align*}45^\circ\end{align*}.

Review

  1. \begin{align*}RACE\end{align*} is a rectangle. Find:
    1. \begin{align*}RG\end{align*}
    2. \begin{align*}AE\end{align*}
    3. \begin{align*}AC\end{align*}
    4. \begin{align*}EC\end{align*}
    5. \begin{align*}m \angle RAC\end{align*}

  2. \begin{align*}DIAM\end{align*} is a rhombus. Find:
    1. \begin{align*}MA\end{align*}
    2. \begin{align*}MI\end{align*}
    3. \begin{align*}DA\end{align*}
    4. \begin{align*}m \angle DIA\end{align*}
    5. \begin{align*}m \angle MOA\end{align*}

  3. \begin{align*}CUBE\end{align*} is a square. Find:
    1. \begin{align*}m \angle UCE\end{align*}
    2. \begin{align*}m \angle EYB\end{align*}
    3. \begin{align*}m \angle UBY\end{align*}
    4. \begin{align*}m \angle UEB\end{align*}

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. 

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Vocabulary

TermDefinition
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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