Parallelogram Classification
Andrew Gloag
Bill Zahner
Dan Greenberg
Jim Sconyers
Lori Jordan
Victor Cifarelli
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Printed: December 20, 2012
AUTHORS
Andrew Gloag
Bill Zahner
Dan Greenberg
Jim Sconyers
Lori Jordan
Victor Cifarelli
EDITOR
Annamaria Farbizio
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C ONCEPT
Concept 1. Parallelogram Classification
1 Parallelogram Classification
Here you’ll learn what properties differentiate the three special parallelograms: rhombuses, rectangles, and squares.
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.
Watch This
MEDIA
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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 6 A ∼
=6 B∼
=6 C∼
= 6 D.
• 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.
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ABCD is a square if and only if 6 A ∼
=6 B∼
=6 C∼
= 6 Dand 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 6 BAD and 6 BCDand BD bisects 6 ABC and 6 ADC, then ABCD is also a
rhombus.
Example A
What typed of parallelogram are the figures below?
a)
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Concept 1. Parallelogram Classification
b)
Answer:
a) All sides are congruent and one angle is 135◦ , 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.
TABLE 1.1:
Properties of a Parallelogram
• SQkER
Properties of a Rhombus
• SQ ∼
= ER ∼
= SE ∼
= QR
Properties of a Rectangle
• m6 SER = m6 SQR =
m6 QSE = m6 QRE = 90◦
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TABLE 1.1: (continued)
Properties of a Parallelogram
• SEkQR
Properties of a Rhombus
Properties of a Rectangle
• SR ⊥ QE
6
•
6
6
•
6
SEQ ∼
= 6 QER ∼
= 6 SQE ∼
=
EQR
• SR ∼
= QE
QSR ∼
=
SRE
• SA ∼
= AR ∼
= QA ∼
= AE
6
RSE ∼
=
6
QRS ∼
=
All the bisected angles are 45◦ .
MEDIA
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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◦ . 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:
a.
b.
c.
d.
e.
4
RG
AE
AC
EC
m6 RAC
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Concept 1. Parallelogram Classification
2. DIAM is a rhombus. Find:
a.
b.
c.
d.
e.
MA
MI
DA
m6 DIA
m6 MOA
3. CUBE is a square. Find:
a.
b.
c.
d.
m6
m6
m6
m6
UCE
EY B
UBY
UEB
For questions 4-15, determine if the quadrilateral is a parallelogram, rectangle, rhombus, square or none.
4.
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5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
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Concept 1. Parallelogram Classification
15.
For questions 16-21 determine if the following are ALWAYS, SOMETIME, or NEVER true. Explain your reasoning.
16.
17.
18.
19.
A rectangle is a rhombus.
A square is a parallelogram.
A parallelogram is regular.
A square is a rectangle.
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