Solution:
Consider that the angles of the triangle are: (a – d) o, ao and (a + d) o.
We know that the sum of the angles of a triangle is 180°. Therefore, we can write:
$ a-d+a+a+d=180{}^\circ $
$ 3a=180{}^\circ $
$ a=180{}^\circ /3 $
$ a=\text{ }{{60}^{o}} $
It is given that the greatest angle is equal to five times of the least angle
Or, we can write that:
Greatest angle / least angle = 5
Upon putting values, we get:
$ \left( a+d \right)/\left( a-d \right)\text{ }=\text{ }5 $
$ \left( 60+d \right)/\left( 60-d \right)\text{ }=\text{ }5 $
Upon cross-multiplication we get,
$ 60\text{ }+\text{ }d\text{ }=\text{ }300\text{ }\text{ }5d $
$ 6d\text{ }=\text{ }240 $
$ d\text{ }=\text{ }240/6 $
$ d=\text{ }40 $
Hence, angles are:
(a – d) ° = 60° – 40° = 20°
a° = 60°
(a + d) ° = 60° + 40° = 100°
Therefore, the angles of triangle in radians will be:
$ \left( 20\text{ }\times \text{ }\pi /180 \right)\text{ }rad\text{ }=\text{ }\pi /9 $
$ \left( 60\text{ }\times \text{ }\pi /180 \right)\text{ }rad\text{ }=\text{ }\pi /3 $
$ \left( 100\text{ }\times \text{ }\pi /180 \right)\text{ }rad\text{ }=\text{ }5\pi /9 $