Angle CAB is congruent to angle ACD because they are alternate interior angles (note the parallel lines AB and DE). So this makes angle CAB to be 50 degrees.
For similar reasoning, angle BCE = angle ABC = 80 degrees.
Focus on the angles that are centered at point C. Those three angles form a straight line so they add to 180 degrees.
(angle ACD) + (angle ACB) + (angle BCE) = 180
(50) + (angle ACB) + (80) = 180
(angle ACB) + 130 = 180
angle ACB = 180-130
angle ACB = 50 degrees
Ultimately, this leads to triangle ABC having the interior angles of A = 50, B = 80 and C = 50.
Because we have exactly two interior angles that are the same measure, this must mean triangle ABC is isosceles. The congruent angles are opposite the congruent sides, meaning that AB = BC.
