angles

Let ∠ABC = x, ∠BCD = 2x , and ∠CDE = 3x. If ∠BAE is 90º, what is the measure of ∠ABC in degrees?

https://web2.0calc.com/questions/please-help_54385

(Answered brilliantly by thelizzybeth, Jun 25, 2020)

arc BD  = 4x 

And angle  DEA will intercept  an   arc of 90 +  BD =   90 + 4x

So...its measure  = (90 +  4x)   / 2

And angle CDE  = 3x

Let BC  and DC  intersect  AE  at   F and G respectively

Angle   FGC   = 180 - [ angle  DEA  + angle CDE ] = 180 -  [(90 + 4x)  /2  +  3x]

Angle  GFC  = 90 - BCA  = 90 - x

Angle BCD  =2x

So  ....in triangle  GFC  we  that

Angle FGC  + Angle  BCD  + Angle  GFC   =180

180 - [ (90 + 4x)  / 2  +  3x ]   + 2x + [   90 - x ]  = 180

180 - 45  - 2x   -  3x  +  2x  +  90  - x  = 180

 -4x  + 45   = 0

4x  = 45

x  = 45/4 =     11.25° = ABC