Draw a circle, a diameter line, and an angle with its ends at the ends of the diameter line and its vertex on the circle. This is your inscribed angle, and it forms a triangle.
Draw a line from the vertex to the center of the circle. This is obviously a radius line, but it also splits the orignal triangle into two smaller triangles. On each of these triangles, two of their sides are radius lines, so the two triangles are isosceles.
Label the angles at the center as "A" and "B". Since a straight line (being the diameter line) has a "measure" of 180 degrees, then m(A + B) = 180 = m(A) + m(B). Whatever the other angles in the one smaller triangle measure, their measures are the same; label them as "C", so the triangle's angles are "A" and "C". Similarly, label the other angles in the other smaller triangle as "D".
We know that m(A) + 2m(C) = m(B) + 2m(D) = 180. Then:
. . . . .m(A) + 2m(C) + m(B) + 2m(D) = 360
By substitution, we get:
. . . . .(180) + 2m(C) + 2m(D) = 360
. . . . .2m(C) + 2m(D) = 180
Divide through by 2 to get:
. . . . .m(C) + m(D) = 90
What does this tell you about the measure of the original inscribed angle?