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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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.
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