Question

Quadrilateral ABCD is inscribed in a circle. Find the measure of each of the angles of the quadrilateral. Show your work.

Answer 1

Your statement that angle C and angle D adding to 180 degrees isn't always true. What is always true is that angle A and angle C add to 180 degrees.

Rule: The opposite angles of an inscribed quadrilateral are supplementary (add to 180 degrees)

So,
(angle A) + (angle C) = 180
(x+2) + (x-2) = 180
x+2+x-2 = 180
2x = 180
2x/2 = 180/2
x = 90

Once you know x, you can find the angles
measure of angle A = x+2
measure of angle A = 90+2
measure of angle A = 92 degrees

measure of angle C = x-2
measure of angle C = 90-2 
measure of angle C = 88 degrees

Note how A+C = 92+88 = 180

measure of angle D = x-10
measure of angle D = 90-10
measure of angle D = 80 degrees

measure of angle B = 360-A-C-D ... see note below
measure of angle B = 360-92-88-80
measure of angle B = 100 degrees

Note: for any quadrilateral, the four angles add to 360 degrees. So A+B+C+D = 360. We can solve for B to get B = 360-A-C-D