Question

For what value of x in the figure points A, B, C, and D are concyclic?

(A) \[9{}^\circ \]
(B) \[10{}^\circ \]
(C) \[11{}^\circ \]
(D) \[12{}^\circ \]

Answer 1

Hint: According to the question, we have a quadrilateral whose all vertices are lying on the circumference of the circle. It means that the quadrilateral ABCD is a cyclic quadrilateral. We know the property that the opposite angle of a cyclic quadrilateral is supplementary. Here, \[\angle B\] and \[\angle D\] are opposite to each other. So, the summation of the measure of \[\angle B\] and \[\angle D\] is \[180{}^\circ \] i.e, \[\angle B+\angle D=180{}^\circ \] . The measure of the \[\angle B\] and \[\angle D\] are \[\left( 81{}^\circ +x \right)\] and \[89{}^\circ \] . Put the measure of \[\angle B\] and \[\angle D\]in the equation, \[\angle B+\angle D=180{}^\circ \]. Now, solve it further and get the value of x.

Complete step-by-step answer:
According to the question, we have a quadrilateral ABCD whose all vertices are lying on the circumference of the circle. The measure of the angles \[\angle B\] and \[\angle D\] is given.
The measure of \[\angle B=81{}^\circ +x\] ………………………………(1)
The measure of \[\angle D=89{}^\circ \] …………………………………(2)
Since a cyclic quadrilateral is a quadrilateral whose all vertices lie on the circumference of the circle and the quadrilateral ABCD has its all vertices on the circumference of the circle so, the quadrilateral ABCD is cyclic.
In the given cyclic quadrilateral ABCD, we have \[\angle B\] and \[\angle D\] opposite to each other.
We know the property of a cyclic quadrilateral that the opposite angles are supplementary. In other words, the summation of the opposite angles is equal to \[180{}^\circ \] .
Using this property, we can say that the summation of the angles, \[\angle B\] and \[\angle D\] is equal to \[180{}^\circ \] . So,
\[\angle B+\angle D=180{}^\circ \] ………………………..(3)
From equation (1) and equation (2), we have the measure of the angles \[\angle B\] and \[\angle D\] that is \[\angle B=81{}^\circ +x\] and \[\angle D=89{}^\circ \] .
Now, putting the measure of \[\angle B\] and \[\angle D\] in equation (3), we get
\[\begin{align}
  & \Rightarrow \angle B+\angle D=180{}^\circ \\
 & \Rightarrow 81{}^\circ +x+89{}^\circ =180{}^\circ \\
\end{align}\]
Solving the above equation further to get the value of x,
\[\begin{align}
  & \Rightarrow x+170{}^\circ =180{}^\circ \\
 & \Rightarrow x=180{}^\circ -170{}^\circ \\
\end{align}\]
\[\Rightarrow x=10{}^\circ \]
Now, from the above equation, we have got the value of x.
Therefore, the value of x is equal to \[10{}^\circ \] .
Hence, the correct option is (B).

Note: We know that the summation of the opposite angles of a cyclic quadrilateral is supplementary. Here, one might make a silly mistake and take the measure of the supplementary angle equal to \[360{}^\circ \] . this is wrong because when two angles are supplementary then their sum is equal to \[180{}^\circ .\]