Quadrilateral Lesson

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Quadrilateral Lesson

• There are seven major types of
• quadrilaterals parallelogram,
• rhombus, rectangle, square, kite,
• trapezoid, and isosceles trapezoid.

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Definition A quadrilateral is a parallelogram
if and only if its opposite sides are parallel.
B
C
A
D
AB CD, BC AD
3
Definition A quadrilateral is a rhombus if and
only if its four sides are equal in length.
E
F
G
H
EF FG GH HE
4
Definition A quadrilateral is a rectangle if
and only if it has four right angles.
J
K
I
L
I, J, K, L are right angles.
5
Definition A quadrilateral is a square if and
only if it has four equal sides and four right
angles.
M
N
O
P
MN NO OP PM
M, N, O, P are right angles.
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From the definitions, one can see that every
square is a rhombus, since every square has four
equal sides. One can also conclude that every
square is a rectangle, since every square has
four right angles. This information is
summarized in the network below. This network
shows a part of a hierarchy of quadrilaterals.
rectangle
rhombus
square
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Because two perpendiculars to the same line are
parallel, every rectangle is a parallelogram. As
a result, parallelogram can be added to the
hierarchy.
B
C
A
D
parallelogram
rhombus
rectangle
square
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A fifth type of quadrilateral is formed by the
union of two isosceles triangles having the same
base, with the base re- moved. The result is a
quadrilateral that resembles a kite or
arrowhead. Pictured here is the convex kite ABCD.
B
A
C
AB BC AD DC
D
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Definition A quadrilateral is a kite if and
only if it has two distinct pairs of consecutive
sides of the same length.
B
A
C
AB BC AD DC
D
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From the definitions of kite and rhombus, one can
conclude that every rhombus is a kite. This
information is added to the hierarchy. One can
now conclude that every square is a kite by
reading up the hierarchy from square to rhombus
to kite.
kite
parallelogram
rectangle
rhombus
square
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Definition A quadrilateral is a trapezoid if
and only if it has at least one pair of parallel
sides.
T
R
TR PA
P
A
Parallel sides of a trapezoid are called bases.
In the figure above, TR and PA are bases. Two
consecutive angles that share a base are called
base angles. This terminology enables the class
to de- fine a special type of trapezoid.
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Definition A trapezoid is isosceles if and only
if it has a pair of base angles equal in
measure.
A
B
isosceles trapezoid with bases AB and CD
C
D
AB DC, m D m C
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Because a rectangle has opposite sides parallel
and all angles equal, every rectangle is an
isosceles trapezoid. You can now relate
all these seven types of quadrilaterals in the
same hierarchy. This is shown by the dark lines
below.
quadrilateral
kite
trapezoid
isosceles trapezoid
parallelogram
rhombus
rectangle
square
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The hierarchy of quadrilaterals is very useful
because it allows properties of some
quadrilaterals to apply to other
quadrilaterals. The general rule is ANY
property held by a type of figure in the
hierarchy is also held by all the types of
figures below it to which it is connected. For
example, square is below rhombus in the
hierarchy. Thus, any square has all the
properties of a rhombus. Squares and rhombuses
are below kite. Thus, they have all the
properties of kites.
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For further assessment - Look at students
note page. - Look at quadrilateral worksheet
page. (Both of these documents may be linked
from the Main Lesson Page)