Mathematics

Find the measure of an angle if its supplement-three times complement $$={20}^{\circ}{30}^{\prime}{40}^{\prime\prime}$$ 


SOLUTION
Let the measure of an angle be $${x}^{\circ}$$

$$\therefore$$ measure of its supplement$$={180}^{\circ}-{x}^{\circ}$$

The measure of its complement$$={90}^{\circ}-{x}^{\circ}$$

$$\Rightarrow \left({180}^{\circ}-{x}^{\circ}\right)-3\left({90}^{\circ}-{x}^{\circ}\right)={20}^{\circ}{30}^{\prime}{40}^{\prime\prime}$$ 

$$\Rightarrow \left({180}^{\circ}-{x}^{\circ}\right)-{270}^{\circ}+3{x}^{\circ}={20}^{\circ}{30}^{\prime}{40}^{\prime\prime}$$ 

$$\Rightarrow 2{x}^{\circ}={20}^{\circ}{30}^{\prime}{40}^{\prime\prime}+{90}^{\circ}$$ 

$$\Rightarrow 2{x}^{\circ}={110}^{\circ}{30}^{\prime}{40}^{\prime\prime}$$ 

$$\Rightarrow {x}^{\circ}=\dfrac{{110}^{\circ}{30}^{\prime}{40}^{\prime\prime}}{2}$$ 

$$\Rightarrow {x}^{\circ}={55}^{\circ}{15}^{\prime}{20}^{\prime\prime}$$

The measure of the angle is $${55}^{\circ}{15}^{\prime}{20}^{\prime\prime}$$
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