Given a trapezoid ABCD with AD parallel to BC. The diagonals AC and BD are congruent. Prove that ABCD is an isosceles trapezoid.

To prove that trapezoid ABCD is isosceles, we need to show that its opposite sides are congruent. In this case, we need to show that AB is congruent to CD.

Since AD is parallel to BC with AC and BD congruent, we have two pairs of congruent triangles: triangle ABC and triangle CDA (by side-angle-side congruence), and triangle BCD and triangle DAB (by side-angle-side congruence).

First, let's consider triangle ABC and triangle CDA. From the given information, we know that AC and BD are congruent. Since AC and BD are diagonals, they intersect at point P. So we have two congruent sides, AC and BD, and the included angle at point C is common to both triangles.

By the SAS (side-angle-side) congruence condition, we can conclude that triangle ABC is congruent to triangle CDA. Therefore, the corresponding angles of these triangles are congruent.

Now let's consider triangle BCD and triangle DAB. Again, we have two congruent sides, AC and BD, and the included angle at point D is common to both triangles.

Using the SAS congruence condition, we can conclude that triangle BCD is congruent to triangle DAB. Therefore, the corresponding angles of these triangles are congruent.

Now, let's compare the corresponding angles of triangles ABC and CDA. The angle at point A in triangle ABC corresponds to the angle at point C in triangle CDA, and the angle at point B in triangle ABC corresponds to the angle at point D in triangle CDA.

Since the corresponding angles of congruent triangles are congruent, we can conclude that angle A = angle C and angle B = angle D.

Now, considering trapezoid ABCD, we have two pairs of congruent opposite angles, A = C and B = D.

Since opposite angles of a trapezoid are supplementary (sum to 180 degrees), we can write the following equation:

A + C = 180 degrees and B + D = 180 degrees

Substituting the congruent angles, we have:

angle A + angle C = 180 degrees and angle B + angle D = 180 degrees

Since angle A = angle C and angle B = angle D, we can rewrite the equation as:

angle A + angle A = 180 degrees and angle B + angle B = 180 degrees

Simplifying, we get:

2 * angle A = 180 degrees and 2 * angle B = 180 degrees

Dividing both sides of each equation by 2, we have:

angle A = 90 degrees and angle B = 90 degrees

Therefore, trapezoid ABCD is an isosceles trapezoid, with AB congruent to CD.