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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.

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)

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 $45^\circ$ .

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.

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 Language: English Spanish

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 $\angle B \cong \angle B$