Let ∠ABC=x, ∠BCD=2x, and ∠CDE=3x. If ∠BAE is 90∘, what is the angle of ∠ABC in degrees?
I say this for every geometry problem I solve, but I really wish I had a diagram. In order for my solution to make sense, please draw out what I wrote below while reading it. :)
Extend line BA to the other side of the circle. Let's call the point where it touches the circle F. Inscribed angle CBF, as given by the problem, is x. So arc CF is 2x because a central angle/arc equals twice the inscribed angle.
Arc CE is 6x because ∠CDE=3x.
Since∠BAE=90, and line BAF is straight, then EAF is also ∠90.
arc CF + arc CE = arc EAF
2x + 6x = 90
8x = 90
x = 11.25
I'm not sure if my solution is correct because I didn't utilize ∠BCD=2x.