One angle of a triangle \[\frac{2}{3}\] x grades and another is \[\frac{3}{2}\] *x* degrees while the third is \[\frac{\pi x}{75}\] radians. Express all the angles in degrees.

#### Solution

One angle of the triangle = \[\frac{2}{3}x \text{ grad }\]

\[= \left( \frac{2}{3}x \times \frac{9}{10} \right)^\circ\left[ \because 1 \text{ grad }= \left( \frac{9}{10} \right)^\circ\right]\]

\[ = \left( \frac{3}{5}x \right)^\circ\]

Another angle = \[\left( \frac{3}{2}x \right)^\circ\]

\[\because 1\text{ radian }= \left( \frac{180}{\pi} \right)^\circ\]

\[\text{ Third angle of the triangle }= \frac{x\pi}{75}\text{ rad }\]

\[ = \left( \frac{180}{\pi} \times \frac{x\pi}{75} \right)^\circ\]

\[ = \left( \frac{12}{5}x \right)^\circ\]

Now,

\[\frac{3}{5}x + \frac{3}{2}x + \frac{12}{5}x = 180 \text{ (Angle sum property) }\]

\[ \Rightarrow \frac{6x + 15x + 24x}{10} = 180\]

\[ \Rightarrow \frac{45x}{10} = 180\]

\[ \Rightarrow x = 40\]

Thus, the angles are:

\[\left( \frac{3}{5}x \right)^\circ= 24^\circ\]

\[\left( \frac{3}{2}x \right)^\circ = 60^\circ \]

\[ \left( \frac{12x}{5} \right)^\circ= 96^\circ\]