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.