Question

​ Quadrilateral ABCD ​ is inscribed in this circle.

What is the measure of angle C?

Enter your answer in the box.


°




Image shows a circle with inscribed quadrilateral A B C D. The measure of angle A is 2 x plus 3 degrees. The measure of angle B is 2 x minus 4 degrees. The measure of angle D is 3 x plus 9 degrees.

Answer 1

When the quadrilateral is inscribed in a circle (a "cyclic" quadrilateral), opposite angles are supplementary. This fact gives a relationship between angles B and D that lets us find the value of x.
∠B + ∠D = 180
(2x-4) + (3x+9) = 180
5x +5 = 180
x = (180 -5)/5 = 35

Now, we can find angle C as
∠C = 180 - ∠A
∠C = 180 - (2×35+3)
∠C = 107°