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 ….

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). The length of an altitude is called the **height** of the trapezoid.

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

*AB *and *CD *are bases.

*XY *is an altitude.

*MN *is the median.

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

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 shows that a parallelogram has two sets of bases and that, with each set of bases, there is an associated height.

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

In ABCD,

*XY *is an altitude to bases *AB *and *CD*

*JK *is an altitude to bases *AD *and *BC*

*XY* is the height of *ABCD*, with *AB *and *CD *as bases, JK is the height of *ABCD*, with *AD *and *BC *as bases.

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.*

- ∠
*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), by *Theorem 45, AE = EC* and *BE = ED.*

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