Question

A quadrilateral ABCD is inscribed in a circle O. What is the measure of $\angle C$ ?

Answer 1

Hint: To find the value of $\angle C$ , we have to find the value of x. We have to apply the property that the sum of opposite angles in a cyclic quadrilateral is equal to $180{}^\circ $ . Hence, the sum of $\angle B$ and $\angle D$ is $180{}^\circ $ . We have to substitute the values of these angles from the figure and solve for x. Then, we have to substitute the value of x in the $\angle C$ from the figure.

Complete step by step solution:
We have to find the value of $\angle C$ . We know that in a cyclic quadrilateral, the sum of opposite angles is equal to $180{}^\circ $ .
$\Rightarrow \angle B+\angle D=180{}^\circ $
From the figure, we can see that $\angle B=2x+3\text{ and }\angle D=4x+3$ .Let us substitute these values in the above equation.
$\Rightarrow 2x+3+4x+3=180{}^\circ $
We have to find the value of x. Let us combine the like terms on the LHS.
$\Rightarrow 2x+4x+3+3=180{}^\circ $
Let us add the like terms on the LHS.
$\Rightarrow 6x+6=180{}^\circ $
Let us collect the constant on the RHS.
$\begin{align}
  & \Rightarrow 6x=180{}^\circ -6 \\
 & \Rightarrow 6x=174{}^\circ \\
\end{align}$
We have to take the coefficient of x to the RHS.
$\Rightarrow x=\dfrac{174{}^\circ }{6}=29{}^\circ $
Now, we are given that $\angle C=2x+1$ . Let us substitute the value of x in this angle.
$\angle C=\left( 2\times 29{}^\circ \right)+1=58{}^\circ +1=59{}^\circ $

Therefore, the measure of $\angle C$ is $59{}^\circ $

Note: Students must know what a cyclic quadrilateral is and their properties. A cyclic quadrilateral is a quadrilateral which has all its four vertices lying on a circle. It is also sometimes called inscribed quadrilateral. They may misunderstand that the opposite angles of a cyclic quadrilateral are equal.