Hypotenuse-Leg triangle congruence criteria can be proved similarly to how SSS was proved. An auxiliary line is drawn from B to E, which creates an isosceles triangle, BAE (blue triangle).[br][br]Why do we know that triangle BCE (orange triangle) is also an isosceles triangle, even though we are not given that BC = EF?
Angles ABC and DEF are right angles. The base angles of the blue triangle (angle ABE or angle DEB) plus the base angles of the orange triangle (angle CBE or angle FEB) together make up that right angle. Thus the base angles of the orange triangle are congruent complements to the base angles of the blue triangle, so angle CBE is congruent to angle FEB, as can be shown by measuring the angles. If the base angles are congruent, the the orange triangle BCE is isosceles.