Question

prove that the opposite sides of a cyclic quadrilateral are supplementary

Answer 1

Construct a radius to each of the four vertices of the quadrilateral 

Since the radii of the circle are all congruent, this partitions the quadrilateral into four isosceles triangles. The base angles of an isosceles triangle have the same measure. 

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.