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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.

image0.jpg
  • 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:

image1.png

Statement 1:

image2.jpg image3.png

Reason for statement 1: Given.

Statement 2:

image4.png

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

Statement 3:

image5.png

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

Statement 4:

image6.png

Reason for statement 4: Reflexive Property.

Statement 5:

image7.png

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

Statement 6:

image8.png

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

Statement 7:

image9.png

Reason for statement 7: If angles, then sides.

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