# Proof - A trapezoid is isosceles if and only if the two base angles are equal.

## Definition of Isoceles Trapeziod

A trapezoid is isosceles if the non-parallel sides are equal in length.

## Isosceles Trapezoid

To prove: a trapezoid is isosceles if and only if the two base angles are equal.

### Part A: If a trapezoid is isosceles, the base angles are equal.

To show: If AE is congruent with DE, then angle A is congruent to angle D.

1. Let ABCD be a trapezoid where BC is parallel to AD.
2. Assume segments AB = CD.
3. Extend the segments AB and CD. Let the point of their intersection be point E. How do we know they intersect?.
4. Since BC is parallel to AD by definition, we can conclude that AE/AB = DE/DC. But, since AB = DC by the definition of an Isoceles Trapezoid, we can conclude that AE = DE.
5. Given that AE = DE we can conclude that the triangle AED is isosceles. We can then further conclude that angle EAD = angle EDA, which was to be shown.

### Part B: If the base angles of a trapezoid are equal, the trapezoid is isosceles.

To show: If angle A is congruent to angle D, then AE is congruent with DE.

1. Let ABCD be a trapezoid where BC is parallel to AD.
2. To be shown: If angle BAD = angle CDA then segment AB = DC.
3. Assume angle BAD = angle CDA.
4. Extend the segments AB and DC until they meet. How do we know they intersect? Call the point of intersection E.
5. Since angle BAD = angle EDA, we can conclude that triangle AED is isosceles.
6. Within the triangle AED, since BC is parallel to AD by definition, we can conclude that AB/AE = DC/DE. But, since AE = DE by the definition of an isosceles triangle, we can conclude that AB = DC, which was to be shown.