If it is true that not all quadrilaterals are created equal, the same may be said about parallelograms. You can even out the sides or stick in a right angle.
Rectangle
A
rectangle is a quadrilateral with all right angles. It is easily shown that it must also be a parallelogram, with all of the associated properties. A rectangle has an additional property, however.
Theorem 51: The diagonals of a rectangle are equal.
In rectangle
ABCD (Figure
1 ),
AC =
BD, by
Theorem 51.
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Figure 1
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The diagonals of a rectangle are equal.
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Rhombus
A
rhombus is a quadrilateral with all equal sides. It is also a parallelogram with all of the associated properties. A rhombus, however, also has additional properties.
Theorem 52: The diagonals of a rhombus bisect opposite angles.
Theorem 53: The diagonals of a rhombus are perpendicular to one another.
In rhombus
CAND (Figure
2 ), by
Theorem 52,
CN
bisects ∠
DCA and ∠
DNA. Also,
AD
bisects ∠
CAN and ∠
CDN and by
Theorem 53,
CN
⊥
AD
.
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Figure 2
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The diagonals of a rhombus are perpendicular to one another and bisect opposite angles.
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Square
A
square is a quadrilateral with all right angles and all equal sides. A square is also a parallelogram, a rectangle, and a rhombus and has all the properties of all these special quadrilaterals. Figure
3 shows a square.
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Figure 3
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A square has four right angles and four equal sides.
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Figure
4 summarizes the relationships of these quadrilaterals to one another.
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Figure 4
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The relationships among the various types of quadrilaterals.
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Example 1: Identify the following figures.
5
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Figure 5
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Identify these polygons.
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(a) pentagon, (b) rectangle, (c) hexagon, (d) parallelogram, (e) triangle, (f) square, (g) rhombus, (h) quadrilateral, (i) octagon, and (j) regular pentagon
Example 2: In Figure
6 , find
m ∠
A, m ∠
C,
m ∠
D,
CD, and
AD.
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Figure 6
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A parallelogram with one angle specified.
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m ∠
A =
m ∠
C = 80°, because consecutive angles of a parallelogram are supplementary.
m ∠
D = 100°, because opposite angles of a parallelogram are equal.
CD = 8 and
AD = 4, because opposite sides of a parallelogram are equal.
Example 3: In Figure
7 , find
TR, QP, PS, TP, and
PR.
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Figure 7
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A rectangle with one diagonal specified.
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TR = 15, because diagonals of a rectangle are equal.
QP =
PS =
TP =
PR = 7.5, because diagonals of a rectangle bisect each other.
Example 4: In Figure
8 , find
m ∠
MOE, m ∠
NOE, and
m ∠
MYO.
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Figure 8
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A rhombus with one angle specified.
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m ∠
MOE =
m ∠
NOE = 70°, because diagonals of a rhombus bisect opposite angles.
m ∠
MYO = 90°, because diagonals of a rhombus are perpendicular.
Fundamental Ideas
Polygons
