Question

Find the value of $x$ in the given figure given below:
 

A. ${118}^{\circ }$
B. ${20}^{\circ }$
C. ${72}^{\circ }$
D. ${223}^{\circ }$

Answer 1

Hint: In this question we have to find the value of $x$ . To find the value of $x$ we will use the sum of all interior angles of a triangle property and straight angle or straight-line definition. By using these properties, we can find the value of $x$ .
The sum of all interior angles of the triangle is ${180}^{\circ }$ .
The angle of the straight line is ${180}^{\circ }$ .

Complete step-by-step answer:
Consider the given figure. Draw a straight line from a point D which bisects the line $AB$ . Let name the point as E which bisects the line $AB$ .
Using the property “the sum of all interior angles of a triangle is ${180}^{\circ }$ ”
Now consider the $△ACE$
The sum of all interior angles is,
 ${48}^{\circ }+{30}^{\circ }+a={180}^{\circ }$
 $\Rightarrow {78}^{\circ }+a={180}^{\circ }$
 $\Rightarrow a={180}^{\circ }-{78}^{\circ }$
 $\Rightarrow a={102}^{\circ }$
Therefore the $|\underset{―}{E}$ in the $△ACE$ is ${102}^{\circ }$ .
      

Now consider the straight line $AB$ where E is the midpoint of $AB$ . We know that the angle of the straight line is ${180}^{\circ }$ . We know value of the $|\underset{―}{AEC}$ is ${102}^{\circ }$ .
Therefore, we have $|\underset{―}{E}=|\underset{―}{AEC}+|\underset{―}{CEB}$
 $\Rightarrow |\underset{―}{E}={102}^{\circ }+y$
 $$\begin{gathered} \Rightarrow {180}^{\circ }={102}^{\circ }+y \\ \Rightarrow y={180}^{\circ }-{102}^{\circ } \end{gathered}$$
 $\Rightarrow y={78}^{\circ }$
Now let us consider the $△DEB$
The sum of all interior angles is,
 ${40}^{\circ }+{78}^{\circ }+z={180}^{\circ }$
 $\Rightarrow {118}^{\circ }+z={180}^{\circ }$
 $\Rightarrow z={180}^{\circ }-{118}^{\circ }$
 $\Rightarrow z={62}^{\circ }$
Therefore $|\underset{―}{EDB}$ IS ${62}^{\circ }$ .
Now consider the straight line $EC$ where D is the midpoint of $EC$ . We know that the angle of the straight line is ${180}^{\circ }$ . We know value of the $|\underset{―}{DEB}$ is ${62}^{\circ }$ .
Therefore, we have $|\underset{―}{D}=|\underset{―}{EDB}+|\underset{―}{CDB}$
 $\Rightarrow |\underset{―}{D}={62}^{\circ }+b$
 $$\begin{gathered} \Rightarrow {180}^{\circ }={62}^{\circ }+b \\ \Rightarrow b={180}^{\circ }-{62}^{\circ } \end{gathered}$$
 $\Rightarrow b={118}^{\circ }$
Therefore, the value of $x$ is ${118}^{\circ }$ .
So, the correct answer is “Option A”.

Note: Candidates should be careful in determining the angles which are unknown. Here use these properties. The sum of all interior angles of the triangle is ${180}^{\circ }$ and the angle of the straight line is ${180}^{\circ }$ . Or we can use another property: the sum of interior angles is equal to the opposite exterior angle.