Theorems Dealing with Trapezoids

Theorems Dealing with Trapezoids

Trapezoid

Definition: A trapezoid is a quadrilateral with exactly one pair of parallel sides.

Trapezoid has only one set of parallel sides.
[The median of a trapezoid is parallel to the bases and equal to one-half the sum of the bases.]

A trapezoid has ONLY ONE set of parallel sides.
When proving a figure is a trapezoid, it is necessary to prove that two sides are parallel and two sides are NOT parallel.
Theorems Dealing with Trapezoids 1The median (also called the mid-segment) of a trapezoid is a segment that connects the midpoint of one leg to the midpoint of the other leg.

Theorem: The median (or mid-segment) of a trapezoid is parallel to each base and its length is one half the sum of the lengths of the bases.
(True for ALL trapezoids.)

Isosceles Trapezoid

Definition: An isosceles trapezoid is a trapezoid with congruent legs.
Theorems Dealing with Trapezoids 2

Properties:

  • Isosceles Trapezoid has only one set of parallel sides
  • base angles congruent
  • legs congruent
  • diagonals congruent
  • opposite angles supplementary

Theorems:

  1. A trapezoid is isosceles if and only if the base angles are congruent.
  2. A trapezoid is isosceles if and only if the diagonals are congruent.
  3. If a trapezoid is isosceles, the opposite angles are supplementary.

Never assume that a trapezoid is isosceles unless you are given (or can prove) that information.