Question

An exterior angle of a triangle is ${{120}^{\circ }}$and one of the interior opposite angles is ${{30}^{\circ }}$. Find the other angles of the triangle.

Answer 1

- Hint: First, we should draw the figure to calculate the angle ${{x}^{\circ }}$ and ${{y}^{\circ }}$. Then, by using the property of the triangle that the exterior angle is the exterior angle is equal to the sum of the two interior opposite angles, we find $\angle x$. Then, by using the other property of the triangle that sum of all the angles of the triangle is ${{180}^{\circ }}$, we find $\angle y$.


Complete step-by-step solution -
In this question, we are supposed to find the remaining two angles of the triangle which drawn as:

Now, from the figure drawn above we need to calculate the angle ${{x}^{\circ }}$ and ${{y}^{\circ }}$.
Now, by using the property of triangles that the exterior angle is the exterior angle is equal to the sum of the two interior opposite angles.
So, by applying the above property we get:
$\angle x+{{30}^{\circ }}={{120}^{\circ }}$
Then, solve the above equation to get the value of $\angle x$ as:
$\begin{align}
  & \angle x={{120}^{\circ }}-{{30}^{\circ }} \\
 & \Rightarrow \angle x={{90}^{\circ }} \\
\end{align}$
So, the $\angle x$ is ${{90}^{\circ }}$.
Now, by using the other property of the triangle, the sum of all the angles of the triangle is ${{180}^{\circ }}$.
So, by using the above stated property, we get:
$\angle x+\angle y+{{30}^{\circ }}={{180}^{\circ }}$
Now, substitute the value of $\angle x$ as ${{90}^{\circ }}$ and calculate the angle y as:
 $\begin{align}
  & {{90}^{\circ }}+\angle y+{{30}^{\circ }}={{180}^{\circ }} \\
 & \Rightarrow \angle y={{180}^{\circ }}-{{90}^{\circ }}-{{30}^{\circ }} \\
 & \Rightarrow \angle y={{60}^{\circ }} \\
\end{align}$
So, $\angle y={{60}^{\circ }}$ is the other angle.
Hence, the other angles of the triangle are ${{60}^{\circ }}$ and ${{90}^{\circ }}$.

Note: Another approach to solve this kind of problem is as given below:
Firstly we can use one more property that linear pairs on the straight line are supplementary.
So, according to the property stated above, we get:
\[\begin{align}
  & \angle y+{{120}^{\circ }}={{180}^{\circ }} \\
 & \Rightarrow \angle y={{180}^{\circ }}-{{120}^{\circ }} \\
 & \Rightarrow \angle y={{60}^{\circ }} \\
\end{align}\]
Which gives the same result as $\angle y={{60}^{\circ }}$.
Then, by using the other property of the triangle that sum of all the angles of the triangle is ${{180}^{\circ }}$, we can get $\angle x$.