Question

One of the exterior angles of a triangle is ${120}^{°}$ and the interior opposite angles are in the ratio $3:5$ Find the angles of the triangle.

Answer 1

Here it is given that

exterior angles of a triangle is ${120}^{°}$ and

And we know that the sum of interior opposite angles are equal to exterior angles of a triangle

The ratio of opposite interior angles =

So, Let the interior angles be $3x$ and $5x$

Then we have
$$\begin{gathered} \Rightarrow 3x+5x=120 \\ \Rightarrow 8x=120 \\ \Rightarrow x=\frac{120}{8} \\ \Rightarrow x=15 \end{gathered}$$ [Exterior angle is equal to the sum of interior opposite angles.]
Then the two interior angles are $3x=3\times 15={45}^{°}$, and $5x=5\times 15={75}^{°}$

Now

The given angles are ${45}^{°},{75}^{°}$ and we have to find the third angle Let the third angle is $y$

We know that sum of all interior angles of a triangle are ${180}^{°}$

So, we have,

$$\begin{gathered} 45+75+y=180 \\ \Rightarrow 120+y=180 \\ \Rightarrow y=180-120 \\ \Rightarrow y=60 \end{gathered}$$

Hence, All the angles of a triangle are ${45}^{°},{75}^{°}$ and ${60}^{°}$