Comment by kingofmen

So, the line BE is just the line CB extended. It's the same line. And we know that the angles of a triangle add up to 180. And we know that the line BD is defined as perpendicular to AB.

That means the angle ABC and angle DBE must add up to 90. But that's also true of the angles ABC and angle CAB. That means that angle DBE and angle CAB must be the same. Both triangles ABC and BDE are both right triangles, so that means angles ABC and BDE are the same. So they're similar triangles: They have all the same angles.

Additionally, the point D is just at a point so that the length of line segment BD and the length of line segment AB are both the same: c. Since we know that the hypotenuse of triangle ABC is c, and the hypotenuse of triangle BDE is also c, and we know they're both similar triangles, then these triangles must be congruent as well.

We know angle EBD equals BAC, since the sum of triangle ABC's interior angles is 180 degrees and the sum of the 3 angles at B are also 180 degrees. We also know angle DEB is 90 degrees since DE was constructed to be perpendicular to CB. Finally, D was placed at a distance c from B. The two triangles have the same angles and the same side lengths opposite the right angles, so they must be congruent.

It wasn't obvious to me either. But we know the angles ABC, ABD, and DBE equal 180 degrees, as do the interior angles of triangle ABC. From that we can deduce that angle BAC = angle DBE, from which it follows that angle ABC = angle BDE.