 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. ${\text{4}}{{\text{5}}^{\text{o}}}$ Verified
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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.

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.