Hey birkoof.
You have an isosceles triangle, a hemi-circle and another circle [in the top].
Try writing down as many variables as possible with your picture and show us the work you have done to write them as formulas so we can give you a hand.
Well, inside the triangle I have 2alpha + beta = 180deg.
Using the Angle in the Center Theorem I get a X that is two times bigger than my beta (beta = x/2)
Implementing that in 2alpha + beta = 180deg i came with the result that alpha is equal to 90deg - x/4.
That's all i can get from the information given.
Thanks, this helped a lot. Now i just express $\displaystyle \beta $ from the first triangle, which is $\displaystyle 180{}^{\circ} - 2 \alpha $ , put in the $\displaystyle \alpha +3\beta =180{}^{\circ}$ and i get that $\displaystyle \alpha = 360{}^{\circ} /5 $ which is $\displaystyle 72{}^{\circ} $ and the right answer. Thanks!