The Properties of Trapezoids and Isosceles Trapezoids

A trapezoid is a quadrilateral with exactly one pair of parallel sides (the parallel sides are called bases). The following figure shows a trapezoid to the left, and an isosceles trapezoid on the right.

• The properties of the trapezoid are as follows:

  • The bases are parallel by definition.

  • Each lower base angle is supplementary to the upper base angle on the same side.

• The properties of the isosceles trapezoid are as follows:

  • The properties of trapezoid apply by definition (parallel bases).

  • The legs are congruent by definition.

  • The lower base angles are congruent.

  • The upper base angles are congruent.

  • Any lower base angle is supplementary to any upper base angle.

  • The diagonals are congruent.

Perhaps the hardest property to spot in both diagrams is the one about supplementary angles. Because of the parallel sides, consecutive angles are same-side interior angles and are thus supplementary. (All the special quadrilaterals except the kite, by the way, contain consecutive supplementary angles.)

Here’s an isosceles trapezoid proof for you:

Statement 1:

Reason for statement 1: Given.

Statement 2:

Reason for statement 2: The legs of an isosceles trapezoid are congruent.

Statement 3:

Reason for statement 3: The upper base angles of an isosceles trapezoid are congruent.

Statement 4:

Reason for statement 4: Reflexive Property.

Statement 5:

Reason for statement 5: SAS, or Side-Angle-Side (2, 3, 4)

Statement 6:

Reason for statement 6: CPCTC (Corresponding Parts of Congruent Triangles are Congruent).

Statement 7:

Reason for statement 7: If angles, then sides.