The ratio of the supplementary and the complementary angles of an angle is 3 : 1. What is the supplementary angle of the given angle?
Show Hint
Let the complement be $C$ and the supplement be $S$.
The difference between the supplement and complement of any angle is always exactly \( 180^\circ - 90^\circ = 90^\circ \).
Since the ratio is $3 : 1$, the difference in ratio units is \( 3 - 1 = 2 \) units.
Therefore, \( 2\text{ units} = 90^\circ \implies 1\text{ unit } (Complement) = 45^\circ \).
So, the Supplement is \( 3\text{ units} = 3 \times 45^\circ = 135^\circ \).
This method allows you to solve the entire problem mentally!
Step 1: Understanding the Concept:
This question focuses on the definitions of complementary and supplementary angles.
Two angles are complementary if their sum is 90\(^\circ\).
Two angles are supplementary if their sum is 180\(^\circ\).
If we let an unknown angle be \( x \), its complement is the amount needed to reach 90\(^\circ\), and its supplement is the amount needed to reach 180\(^\circ\).
The problem provides a relationship between these two values in the form of a ratio, which we can solve using basic algebra. Step 2: Key Formula or Approach:
1. Let the angle be \( x \).
2. Complement \( = 90^\circ - x \).
3. Supplement \( = 180^\circ - x \).
4. Set up the ratio: \( \frac{180 - x}{90 - x} = \frac{3}{1} \). Step 2: Detailed Explanation:
Assume the given unknown angle is \( x \).
According to geometric definitions:
Its complementary angle is \( (90 - x) \) degrees.
Its supplementary angle is \( (180 - x) \) degrees.
The ratio given in the question is Supplement : Complement \( = 3 : 1 \).
Translating this into an algebraic equation:
\[ \frac{180 - x}{90 - x} = \frac{3}{1} \]
Apply cross-multiplication to solve for \( x \):
\[ 1 \times (180 - x) = 3 \times (90 - x) \]
\[ 180 - x = 270 - 3x \]
Collect the \( x \) terms on one side and constant values on the other:
\[ 3x - x = 270 - 180 \]
\[ 2x = 90 \]
Dividing by 2 gives the value of the original angle:
\[ x = 45^\circ \]
The question asks for the "supplementary angle of the given angle."
Now that we know \( x = 45^\circ \), we find its supplement:
\[ \text{Supplementary Angle} = 180^\circ - 45^\circ = 135^\circ. \]
Check the logic: The complement of 45\(^\circ\) is 45\(^\circ\). The ratio of 135 to 45 is indeed 3 : 1. Step 3: Final Answer:
The supplementary angle is 135\(^\circ\).
This matches Option (D).