In the figure ABCD, AB = BC and ∠BAD and ∠BCD are right angles. Find the…

Two right triangles are congruent by the RHS (Right angle-Hypotenuse-Side) rule when their hypotenuses are equal and one pair of corresponding legs is equal; congruent triangles then have every corresponding side and angle equal, including the two sides that are not part of the equal leg pair.

1. Join diagonal BD, which splits quadrilateral ABCD into triangle ABD and triangle CBD.

2. In triangle ABD, ∠BAD = 90°, so BD is the hypotenuse and AB, AD are the two legs.

3. In triangle CBD, ∠BCD = 90°, so BD is the hypotenuse and CB, CD are the two legs.

4. BD is the same segment in both triangles, and AB = BC is given, so the hypotenuse and one leg of triangle ABD match the hypotenuse and one leg of triangle CBD.

5. By RHS congruence, triangle ABD is congruent to triangle CBD, so their remaining corresponding sides are equal: AD = CD.

As a check, the same congruence also gives ∠ABD = ∠CBD and ∠ADB = ∠CDB, so diagonal BD bisects both ∠ABC and ∠ADC -- quadrilateral ABCD is symmetric about diagonal BD, not about diagonal AC, which is consistent with AD = CD rather than any of the unequal or doubled relations.