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Parallelogram Classification
Bill Zahner
Lori Jordan
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Printed: May 15, 2016
AUTHORS
Bill Zahner
Lori Jordan
www.ck12.org
C HAPTER
Chapter 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 designing a patio for you backyard? You decide to mark it off using your tape measure. Two sides
are 21 feet long and two sides are 28 feet long. Explain how you would only use the tape measure to make your
patio a rectangle. After completing this Concept, you’ll be able to answer this question using properties of special
parallelograms.
Watch This
MEDIA
Click image to the left or use the URL below.
URL: https://www.ck12.org/flx/render/embeddedobject/136951
CK-12 Foundation: Chapter6ParallelogramClassificationA
Watch the video at this link to learn more about rectangle and square properties.
Follow this link to watch a video about rhombus properties.
Guidance
Rectangles, rhombuses (the plural is also rhombi) and squares are all more specific versions of parallelograms.
Rectangle Theorem: A quadrilateral is a rectangle if and only if it has four right (congruent) angles.
Rhombus Theorem: A quadrilateral is a rhombus if and only if it has four congruent sides.
Square Theorem: A quadrilateral is a square if and only if it has four right angles and four congruent sides.
From the Square Theorem, we can also conclude that a square is a rectangle and a rhombus.
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Recall that diagonals in a parallelogram bisect each other. Therefore, the diagonals of a rectangle, square and
rhombus also bisect each other. The diagonals of these parallelograms also have additional properties.
Investigation: Drawing a Rectangle
Tools Needed: pencil, paper, protractor, ruler
1. Draw two lines on either side of your ruler, to ensure they are parallel. Make these lines 3 inches long.
2. Remove the ruler and mark two 90◦ angles, 2.5 inches apart on the bottom line drawn in Step 1. Then, draw
the angles to intersect the top line. This will ensure that all four angles are 90◦ . Depending on your ruler, the
sides should be 2.5 inches and 1 inch.
3. Draw in the diagonals and measure them. What do you discover?
Theorem: A parallelogram is a rectangle if and only if the diagonals are congruent.
Investigation: Drawing a Rhombus
Tools Needed: pencil, paper, protractor, ruler
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Chapter 1. Parallelogram Classification
1. Draw two lines on either side of your ruler, to ensure they are parallel. Make these lines 3 inches long.
2. Remove the ruler and mark a 50◦ angle, at the left end of the bottom line drawn in Step 1. Draw the other side
of the angle and make sure it intersects the top line. Measure the length of this side.
3. The measure of the diagonal (red) side should be about 1.3 inches (if your ruler is 1 inch wide). Mark this
length on the bottom line and the top line from the point of intersection with the 50◦ angle. Draw in the fourth
side. It will connect the two endpoints of these lengths.
4. By the way we drew this parallelogram; it is a rhombus because all four sides are 1.3 inches long. Draw in the
diagonals.
Measure the angles created by the diagonals: the angles at their point of intersection and the angles created by the
sides and each diagonal. You should find the measure of 12 angles total. What do you discover?
Theorem: A parallelogram is a rhombus if and only if the diagonals are perpendicular.
Theorem: A parallelogram is a rhombus if and only if the diagonals bisect each angle.
We know that a square is a rhombus and a rectangle. So, the diagonals of a square have the properties of a rhombus
and a rectangle.
Example A
What type of parallelogram are the ones below?
a)
b)
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Answers:
a) All sides are congruent and one angle is 135◦ , meaning that the angles are not congruent. By the Rhombus
Theorem, this is a rhombus.
b) This quadrilateral has four congruent angles and all the sides are not congruent. By the Rectangle Theorem, this
is a rectangle.
Example B
Is a rhombus SOMETIMES, ALWAYS, or NEVER a square? Explain your reasoning.
A rhombus has four congruent sides, while a square has four congruent sides and angles. Therefore, a rhombus is
only a square when it also has congruent angles. So, 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 Parallelograms
• SQ || ER
Properties of Rhombuses
• SQ ∼
= ER ∼
= SE ∼
= QR
Properties of Rectangles
6
•
6
• SE || QR
• SR ⊥ QE
• SQ ∼
= ER
•
6
6
4
SEQ ∼
= 6 QER ∼
= 6 SQE ∼
=
EQR
SER ∼
=
QRE
6
SQR ∼
=
6
• SR ∼
= QE
• SA ∼
= AR ∼
= QA ∼
= AE
QSE ∼
=
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Chapter 1. Parallelogram Classification
TABLE 1.1: (continued)
Properties of Parallelograms
• SE ∼
= QR
Properties of Rhombuses
6
•
6
QSR ∼
=
SRE
6
RSE ∼
=
6
Properties of Rectangles
QRS ∼
=
• SA ∼
= AR
• QA ∼
= AE
6
•
•
6
SER ∼
= 6 SQR
QSE ∼
= 6 QRE
Watch this video for help with the Examples above.
MEDIA
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URL: https://www.ck12.org/flx/render/embeddedobject/136952
CK-12 Foundation: Chapter6ParallelogramClassificationB
Concept Problem Revisited
In order for the patio to be a rectangle, first the opposite sides must be congruent. So, two sides are 21ft and two
are 28 ft. To ensure that the parallelogram is a rectangle without measuring the angles, the diagonals must be equal.
You can find the length of the diagonals by using the Pythagorean Theorem.
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d 2 = 212 + 282 = 441 + 784 = 1225
√
d = 1225 = 35 f t
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.
Interactive Practice
MEDIA
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URL: https://www.ck12.org/flx/render/embeddedobject/113036
Explore More
1. RACE is a rectangle. Find:
a.
b.
c.
d.
e.
RG
AE
AC
EC
m6 RAC
2. DIAM is a rhombus. Find:
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a.
b.
c.
d.
e.
Chapter 1. Parallelogram Classification
MA
MI
DA
m6 DIA
m6 MOA
3. Draw a square and label it CUBE. Mark the point of intersection of the diagonals Y . Find:
a.
b.
c.
d.
m6
m6
m6
m6
UCE
EY B
UBY
UEB
For questions 4-12, determine if the quadrilateral is a parallelogram, rectangle, rhombus, square or none. Explain
your reasoning.
4.
5.
6.
7.
8.
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9.
10.
11.
12.
For problems 13-15, find the value of each variable in the figures.
13.
14.
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Chapter 1. Parallelogram Classification
15.
For questions 16-19 determine if the following are ALWAYS, SOMETIMES, 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.
Answers for Explore More Problems
To view the Explore More answers, open this PDF file and look for section 6.5.
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