Properties of Special Parallelograms

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 ), AC = BD, by Theorem 51.

Figure 1 The diagonals of a rectangle are equal.

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 .

Figure 2 The diagonals of a rhombus are perpendicular to one another and bisect opposite angles.

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.

Figure 3 A square has four right angles and four equal sides.

Figure 4 summarizes the relationships of these quadrilaterals to one another.

Figure 4 The relationships among the various types of quadrilaterals.

Example 1: Identify the following figures 5.

Figure 5 Identify these polygons.

(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.

Figure 6 A parallelogram with one angle specified.

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.

Figure 7 A rectangle with one diagonal specified.

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.

Figure 8 A rhombus with one angle specified.

m ∠ MOE = m ∠ NOE = 70°, because diagonals of a rhombus bisect opposite angles.

m ∠ MYO = 90°, because diagonals of a rhombus are perpendicular.

Geometry