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. \[{\text{4}}{{\text{5}}^{\text{o}}}\]

VerifiedVerified
146.1k+ views
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.