What is wrong in my solution attempt to the $80^\circ$-$80^\circ$-$20^\circ$ problem?

Question

I'm trying to solve the following famous problem:

Here's my attempt, for some reason it leads to a contradiction, where have I gone wrong?

Please do not give me any hints on how to solve the problem on a whole, as I want to solve it on my own. If I'm wrong I'll try another approach, but I'd like to know where I have went wrong.

Answer 1

Angles B and C equal 80 degrees each. with angle A equaling 40 degrees, the sum of degrees is 200, not 180.

Answer 2

The angle at the top of the triangle is not $40^\circ$.