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:
Reason for statement 1: Given.
Reason for statement 2: The legs of an isosceles trapezoid are congruent.
Reason for statement 3: The upper base angles of an isosceles trapezoid are congruent.
Reason for statement 4: Reflexive Property.
Reason for statement 5: SAS, or Side-Angle-Side (2, 3, 4)
Reason for statement 6: CPCTC (Corresponding Parts of Congruent Triangles are Congruent).
Reason for statement 7: If angles, then sides.