How can I prove that an angle in a semi-circle is always 90 degrees?

If we take the diameter of a circle and create an angle on the circumference at point C of the circle from the two points where the diameter meets the circumference (points A and B), the angle created will always equal 90 degrees. To prove this we can draw a line from point C to the centre (point O). We have now created two isosceles triangles (O,A,C) and (O,B,C). Therefore, angle OAC = angle OCA (we will call this angle x) and angle OBC = OBA (we will call this angle y).Our angle at point C, therefore is equal to x+y.We can now return to the original triangle (A,B,C) and using our triangle knowledge we can say:x+y+(x+y)=1802x+2y=180x+y=90