Question

Find the value of the unknown interior angle x in the following figure.

Answer 1

Hint: Use the property that the exterior angle of a triangle is equal to the sum of the opposite interior angles of a triangle. So, for the above figure, $\mathrm{\angle }ADC$ is one of the exterior angles of the triangle given and $\mathrm{\angle }ABC,\mathrm{\angle }ACB$ are its opposite interior angle. So, form the mathematical equation and solve to get the value of x.

Complete step-by-step answer:
Let us start the solution to the above question by drawing the diagram given in the question.

We know that the exterior angle of a triangle is equal to the sum of the opposite interior angles of a triangle. In the above figure, $\mathrm{\angle }ADC$ is one of the exterior angles of the triangle given and $\mathrm{\angle }ABC,\mathrm{\angle }ACB$ are its opposite interior angle, so we can say that $\mathrm{\angle }ADC$ is the sum of $\mathrm{\angle }ABC\text{ and }\mathrm{\angle }ACB$ . Also, it is given in the question that $\mathrm{\angle }ADC=80{}^{\circ }$ , $\mathrm{\angle }ABC=30{}^{\circ }\text{ and }\mathrm{\angle }ACB=x$ . So, if we represent this mathematically, we get
$\mathrm{\angle }ADC=\mathrm{\angle }ABC\text{ + }\mathrm{\angle }ACB$
$\Rightarrow 80=30+x$
If we take 30 to the other side of the equation, the sign changes. So, we get
$80-30=x$
$x=50$
Therefore, we can conclude that the measure of x is equal to $50{}^{\circ }$ .

Note:Generally, it is seen that whenever a student is asked about the exterior angle of the triangle, they misinterpret it as the corresponding interior angle subtracted from $360{}^{\circ }$ , while the correct concept is the corresponding interior angle subtracted from $180{}^{\circ }$ . If you want you can solve the above question by using the angle sum property of a triangle followed by the property of a linear pair of angles, but that would be lengthier than the above process.