We are given a cyclic quadrilateral $ABCD$ inscribed in a circle. The angles are given as follows: $\angle A = 14x$, $\angle B = 16x$, $\angle C = 22x$, and $\angle D = 20x$. We need to find the value of $x$ and the measure of angle $C$.

GeometryCyclic QuadrilateralAnglesSupplementary AnglesCircle Geometry
2025/3/28

1. Problem Description

We are given a cyclic quadrilateral ABCDABCD inscribed in a circle. The angles are given as follows: A=14x\angle A = 14x, B=16x\angle B = 16x, C=22x\angle C = 22x, and D=20x\angle D = 20x. We need to find the value of xx and the measure of angle CC.

2. Solution Steps

Since the quadrilateral ABCDABCD is cyclic, opposite angles are supplementary. This means that the sum of opposite angles is 180180^\circ.
Thus, we have:
A+C=180\angle A + \angle C = 180^\circ
B+D=180\angle B + \angle D = 180^\circ
Substituting the given expressions for the angles:
14x+22x=18014x + 22x = 180
36x=18036x = 180
x=18036x = \frac{180}{36}
x=5x = 5
We can also check the other pair of angles:
16x+20x=18016x + 20x = 180
36x=18036x = 180
x=18036x = \frac{180}{36}
x=5x = 5
Now, we can find the measure of angle CC:
C=22x\angle C = 22x
C=22(5)\angle C = 22(5)
C=110\angle C = 110

3. Final Answer

x=5x = 5
C=110\angle C = 110^\circ

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