Question

prove that an angles in a semicircle is one 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 Alpha and
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°