One angle of a triangle 2 3 x grades and another is 3 2 x degrees… - Vidyadip

Question

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

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Answer

Let $\theta_{1}$ $\theta_{2}$and $\theta_{3}$ be angles of a right angles triangle.
$\theta_{1}=\frac{2}{3}\times\text{gradiants}$
$\theta_{2}=\frac{3}{2}\times\text{degrees}$
$\theta_{2}=\frac{\pi\text{x}}{75}\times\text{radians}$
Now,
We have to express all the angles in degrees,
$\theta_{1}=\Big(\frac{3}{2}\text{x}\times\frac{90}{100}\Big)^{\circ}$
$=\frac{3}{5}\text{x}$
$\theta_{2}=\frac{\pi\text{x}}{75}\times\frac{180}{\pi}$
$=\frac{12\text{x}}{5}$
By angles property,
$\theta_{1}+\theta_{2}+\theta_{3}=180^{\circ}$
$\frac{3}{5}\text{x}^{\circ}+\frac{3}{2}\text{x}^{\circ}+\frac{12\text{x}}{5}=180^{\circ}$
$\Rightarrow\frac{9}{2}\text{x}^{\circ}=180^{\circ}$
$\Rightarrow \text{x}=40^{\circ}$
$\therefore\ \theta_{1}=24^{\circ},\theta_{2}=60^{\circ},\theta_{3}=96^{\circ}$

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