Question

prove that opposite angle of cyclic quadrilateral are supplementary

Answer 1

the sum of one angle on a cyclic quadrilateral and its opposite angle is 180°

Answer 2

The sum of the angles around the center of the circle is 360 degrees. The sum of the angles in each of the triangles is 180 degrees. So if we add up the labeled angle measures and the angles forming the circle around the center center, we get:

2a+2b+2c+2d+360=4⋅180.If we subtract 360 from both sides, we get2a+2b+2c+2d=360ora+b+c+d=180.Alternatively, we may use the fact that the sum of angles in a quadrilateral is 360 degrees and bypass the argument with the angles making a circle.

Note that (a+b) and (c+d) are the measures of opposite angles, and we can simply group the measures in the last equation like this:

(a+b)+(c+d)=180.Likewise, (a+d) and (b+c) are the measures of opposite angles, and we can just rearrange the equation to see that(a+d)+(b+c)=180.So, indeed, we see that the opposite angles in a cyclic quadrilateral are supplementary.

Hope This Helps :)