Question

Find the value of the unknown exterior angle $x$ in the following diagram.

Answer 1

Hint: First of all find the third unknown angle of the triangle by using the condition that the sum of all internal angles of a triangle is ${180}^{\circ }$. Then, the sum of this unknown angle and exterior angle $x$ is ${180}^{\circ }$ because these two angles are supplementary as it is evident from the figure. Use this condition to find the required value.

Complete step-by-step answer:

Let the vertices of the triangle are A, B and C as shown in the above figure.
Two of the triangle’s angles are already given in the question. So we have:
$\Rightarrow <A={60}^{\circ }$ and $<B={60}^{\circ }$
According to the triangle's law, we know that the sum of all the internal angles of a triangle is ${180}^{\circ }$. Applying this law for the above triangle, we’ll get:
$\Rightarrow <A+<B+<C={180}^{\circ }$
Putting $<A={60}^{\circ }$ and $<B={60}^{\circ }$, we’ll get:
$$\begin{gathered} \Rightarrow {60}^{\circ }+{60}^{\circ }+<C={180}^{\circ } \\ \Rightarrow <C={180}^{\circ }-{120}^{\circ }={60}^{\circ }.....\text{(1)} \end{gathered}$$
Thus the third internal angle of the triangle is also $${60}^{\circ }$$.
Further, from the figure, we can say that the sum of the internal angle C and the external angle $x$ is ${180}^{\circ }$ because these two angles are supplementary i.e. lying on the same side of a straight line. So we have:
$\Rightarrow <C+x={180}^{\circ }$
Putting the value of $$<C={60}^{\circ }$$ from equation (1), we’ll get:
$$\begin{gathered} \Rightarrow {60}^{\circ }+x={180}^{\circ } \\ \Rightarrow x={180}^{\circ }-{60}^{\circ } \\ \Rightarrow x={120}^{\circ } \end{gathered}$$

Therefore, the value of exterior angle $x$ is ${120}^{\circ }$.

Note: When all the angles of a triangle are equal then the triangle is called equilateral triangle. The measure of each angle in such a triangle is $${60}^{\circ }$$. Thus the triangle in the above question is an equilateral triangle.
When only two angles of a triangle are the same then the triangle is called isosceles triangle. And when all the angles are different then it is called scalene triangle.