Question

prove that an angle inscribed in a semicircle is a right angle

Answer 1

Proof : 
Label the diameter endpoints A and B, the top point C and the middle of the circle M.
Label the acute angles at A and B Alphaand Beta.
Draw a radius 'r' from the (right) angle point C to the middle M.

Angle MAC = ACM = Alpha because the left subtriangle is iscosceles because the opposite sides AM and CM are both radii.

Angle MBC = BCM = Beta because the right subtriangle is iscosceles because the opposite sides BM and CM are both radii.

Add up the angles at A, B and C.

This gives 2 * Alpha + 2 * Beta, which sum to 180° because ABC is a triangle.

Halving this gives Alpha + Beta (= the angle ACB) = 90°