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# If the supplement of an angle is three times it’s complement, then angle is A. ${\text{4}}{{\text{0}}^{\text{o}}}$B. ${\text{3}}{{\text{5}}^{\text{o}}}$C. ${\text{5}}{{\text{0}}^{\text{o}}}$D. ${\text{4}}{{\text{5}}^{\text{o}}}$

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Hint: As we know that the Supplementary angles are two angles whose sum is ${\text{18}}{{\text{0}}^{\text{o}}}$ while complementary angles are two angles whose sum is ${\text{9}}{{\text{0}}^{\text{o}}}$ .Using the information that supplementary is three times the complementary angle. We form the equation and solve the following for the required angle.

Complete step by step answer:

Using the given data that supplementary angle is three times its the complementary angle.
Let the required angle be x,
As, the Supplementary angles are two angles whose sum is ${\text{18}}{{\text{0}}^{\text{o}}}$ while complementary angles are two angles whose sum is ${\text{9}}{{\text{0}}^{\text{o}}}$ .
So, it’s complementary angle can be given as ${\text{9}}{{\text{0}}^{\text{o}}}{\text{ - x}}$
Let the supplementary angle be ${\text{18}}{{\text{0}}^{\text{o}}}{\text{ - x}}$
Now, using the data that supplementary angle is three times it’s complementary angle, we get,
${\text{18}}{{\text{0}}^{\text{o}}}{\text{ - x = 3(9}}{{\text{0}}^{\text{o}}}{\text{ - x)}}$
On simplifying we get,
$\Rightarrow {\text{18}}{{\text{0}}^{\text{o}}}{\text{ - x = 27}}{{\text{0}}^{\text{o}}}{\text{ - 3x}}$
On rearranging we get,
$\Rightarrow {\text{3x - x = 27}}{{\text{0}}^{\text{o}}}{\text{ - 18}}{{\text{0}}^{\text{o}}}$
On simplification we get,
$\Rightarrow {\text{2x = 9}}{{\text{0}}^{\text{o}}} \\ \Rightarrow {\text{x = }}\dfrac{{{\text{9}}{{\text{0}}^{\text{o}}}}}{{\text{2}}} \\ \Rightarrow {\text{x = 4}}{{\text{5}}^{\text{o}}} \\$
Hence, option (d) is the correct answer.

Note: Two angles are called complementary when their measures add to ${\text{9}}{{\text{0}}^{\text{o}}}$. Two angles are called supplementary when their measures add up to ${\text{18}}{{\text{0}}^{\text{o}}}$. In-plane geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Angles formed by two rays lie in a plane, but this plane does not have to be a Euclidean plane.