# 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.