Diameter of circle subtends 1 rt. angle

Any triangle drawn so all vertices are on a circumference of a circle and one side is the diameter of the same circle, then the angle opposite the diameter is a right angle.

Let AC pass through the centre O of a circle.

Let B be any point on the circumference.

Join AB, CB and BO.

AO = BO = CO                 (radius of circle)

^OBA = ^OAB                (isosceles triangle)

^OBC = ^OCB                (isosceles triangle)

Consider tri. ABC

^OAB + ^OBA + ^OBC + ^OCB  = 2 rt. ang.   (sum of angles in tri.= 2 rt ang.)

Subst. ^OAB = ^OBA and ^OBC = ^OCB 

2(^OBA + ^OBC) = 2 rt. ang.

^OBA + ^OBC = 1 rt.angle

But ^OBA + ^OBC = ^ABC

So

        ^ABC=1 rt. angle