Unlike humans, all quadrilaterals are not created equal. It's not a matter of size I'm alluding to here, but rather a question of features. They may have a pair of parallel sides, two pairs, a right angle ….

## Trapezoid

A **trapezoid** is a quadrilateral with only one pair of opposite sides parallel. The parallel sides are called **bases,** and the *non*parallel sides are called *legs*. A segment that joins the midpoints of the legs is called the **median of the trapezoid.** Any segment that is perpendicular to both bases is called an **altitude** **of the trapezoid** (Figure 1**height** of the trapezoid.

**Figure 1** A trapezoid with its median and an altitude.

* AB *and

*are bases.*CD

* XY *is an altitude.

* MN *is the median.

*XY*, length of segment * XY *is the height.

## Parallelogram

A **parallelogram** is any quadrilateral with both pairs of opposite sides parallel. Each pair of parallel sides is called a pair of **bases of the parallelogram.** Any perpendicular segment between a pair of bases is called the an **altitude of the parallelogram.** The length of an altitude is the height of the parallelogram. The symbol is used for the word parallelogram. Figure

**Figure 2 **A parallelogram with its bases and associated heights.

In ABCD,

* XY *is an altitude to bases

*and*AB

CD

* JK *is an altitude to bases

*and*AD

BC

*XY* is the height of *ABCD*, with * AB *and

*as bases, JK is the height of*CD

*ABCD*, with

*and*AD

*as bases.*BC

The following are theorems regarding parallelograms:

*Theorem 41:* A diagonal of a parallelogram divides it into two congruent triangles.

In *ABCD* with diagonal * BD *according to

*Theorem 41, ΔABD ≅ Δ CDB*(Figure 3

**Figure 3 **Two congruent triangles created by a diagonal of a parallelogram.

*Theorem 42:* Opposite sides of a parallelogram are congruent.

*Theorem 43*: Opposite angles of a parallelogram are congruent.

*Theorem 44:* Consecutive angles of a parallelogram are supplementary.

In *ABCD* (Figure 4

**Figure 4** A parallelogram.

- By
*Theorem 42,**AB*=*DC*and*AD*=*BC.*

- By
*Theorem 43, m*∠*A = m*∠*C*and*m*∠*B = m*∠*D.*

- By
*Theorem 44:*

- ∠
*A*and ∠*B*are supplementary.

- ∠
*B*and ∠*C*are supplementary.

- ∠
*C*and ∠*D*are supplementary.

- ∠
*A*and ∠D are supplementary.

*Theorem 45:* The diagonals of a parallelogram bisect each other.

In *ABCD* (Figure 5*Theorem 45, AE = EC* and *BE = ED.*

**Figure 5 ***The diagonals of a parallelogram bisect one another.*