Question

In a triangle \[\Delta ABC\] if \[\angle A={{42}^{\circ }}\] and \[\angle B=\dfrac{\pi }{6}\] then the measure of \[\angle C\] in degrees is
(a) \[{{106}^{\circ }}\]
(b) \[{{109}^{\circ }}\]
(c) \[{{108}^{\circ }}\]
(d) \[{{107}^{\circ }}\]

Answer 1

Hint: Let us use a rough figure of a given problem.

We solve this problem by converting the given angles to a degree system. We have the formula of conversion of radian system to degree system that is
\[\dfrac{\pi }{2}={{90}^{\circ }}\]
Then we use the standard result that is the sum of angles of a triangle equal to \[{{180}^{\circ }}\] that is for a triangle \[\Delta ABC\] we have
\[\angle A+\angle B+\angle C={{180}^{\circ }}\]
Then, we find the required angle.

Complete step-by-step solution
We are given that for a triangle \[\Delta ABC\] the measure of two angles as
\[\angle A={{42}^{\circ }}\]
\[\angle B=\dfrac{\pi }{6}\]
We are asked to find the other angle in degrees.
So, let us convert the angle of the radian system to the degree system.
We know that the conversion of radian system to degree system that is
\[\dfrac{\pi }{2}={{90}^{\circ }}\]
Now by using the above condition to the given angle we get
\[\begin{align}
  & \Rightarrow \angle B=\dfrac{1}{3}\left( \dfrac{\pi }{2} \right) \\
 & \Rightarrow \angle B=\dfrac{1}{3}\left( {{90}^{\circ }} \right)={{30}^{\circ }} \\
\end{align}\]
Now, let us find the third angle which is required.
We know that the standard result that is the sum of angles of a triangle equal to \[{{180}^{\circ }}\]
By using the above result to \[\Delta ABC\] we get
\[\Rightarrow \angle A+\angle B+\angle C={{180}^{\circ }}\]
Now, by substituting the required values in above equation we get
\[\begin{align}
  & \Rightarrow {{42}^{\circ }}+{{30}^{\circ }}+\angle C={{180}^{\circ }} \\
 & \Rightarrow \angle C={{180}^{\circ }}-{{72}^{\circ }} \\
 & \Rightarrow \angle C={{108}^{\circ }} \\
\end{align}\]
Therefore, the measure of the third unknown angle is given as
\[\therefore \angle C={{108}^{\circ }}\]
So, option (c) is the correct answer.

Note: Students may make mistakes in the addition of angles.
We can add or subtract angles when the angles are in the same system that is either in the radian or degree system. So, we need to convert the angles either into radian system are degree system by using the conversion that is
\[\dfrac{\pi }{2}={{90}^{\circ }}\]
Then we can add or subtract angles.