One angle of a triangle
Question:

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 $=\frac{2}{3} x \mathrm{grad}$

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

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

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

$\because 1 \operatorname{radian}=\left(\frac{180}{\pi}\right)^{\circ}$

Third angle of the triangle $=\frac{x \pi}{75} \mathrm{rad}$

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

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

Now,

$\frac{3}{5} x+\frac{3}{2} x+\frac{12}{5} x=180 \quad$ (Angle sum property)

$\Rightarrow \frac{6 \mathrm{x}+15 \mathrm{x}+24 \mathrm{x}}{10}=180$

$\Rightarrow \frac{45 \mathrm{x}}{10}=180$

$\Rightarrow \mathrm{x}=40$

Thus, the angles are:

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

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

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

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