Comment by da_chicken
Comment by da_chicken 4 days ago
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.
Thank you! Rephrasing for my own understanding: The point of attack that I was missing was that angles BDE and ABC are equal, and now we have two equal angles (which immediately gives us the third) and one equal side, so we're good to go.