Question

The ratio of the supplementary and the complementary angles of an angle is 3 : 1. What is the supplementary angle of the given angle?

• A. 45$^{\circ}$
• B. 60$^{\circ}$
• C. 120$^{\circ}$
• D. 135$^{\circ}$

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!

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
This problem requires an understanding of basic geometric definitions of angles.
Specifically, we need to know the definitions of complementary angles and supplementary angles.
Complementary angles are two angles whose sum is $90^\circ$, whereas supplementary angles are two angles whose sum is $180^\circ$.
By setting up a ratio of these two quantities for a common angle, we can solve for the angle itself and then find its supplementary counterpart.

Step 2: Key Formula or Approach:

• Let the unknown angle be represented as $x$.
  
• The complementary angle of $x$ is defined as: \( 90^\circ - x \).
  
• The supplementary angle of $x$ is defined as: \( 180^\circ - x \).
  
• The ratio of the supplementary angle to the complementary angle is given as:
  \[ \frac{180^\circ - x}{90^\circ - x} = \frac{3}{1} \]
  

Step 3: Detailed Explanation:

• Let us represent the given unknown angle as $x$ (in degrees).
  
• Express the supplementary angle as $(180 - x)$ and the complementary angle as $(90 - x)$.
  
• Set up the algebraic equation based on the given ratio of $3 : 1$:
  \[ \frac{180 - x}{90 - x} = \frac{3}{1} \]
  
• Perform cross-multiplication to solve the linear equation:
  \[ 1 \times (180 - x) = 3 \times (90 - x) \]
  \[ 180 - x = 270 - 3x \]
  
• Rearrange the terms to group the variable $x$ on one side and the constants on the other:
  \[ 3x - x = 270 - 180 \]
  \[ 2x = 90 \]
  
• Divide by 2 to find the value of $x$:
  \[ x = 45^\circ \]
  
• Thus, the original angle is $45^\circ$.
  
• Now, the question asks for the supplementary angle of this given angle $x$.
  
• Calculate the supplementary angle:
  \[ \text{Supplementary Angle} = 180^\circ - x = 180^\circ - 45^\circ = 135^\circ \]
  
• Hence, the supplementary angle is $135^\circ$.
  

Step 4: Final Answer:
The supplementary angle of the given angle is $135^\circ$, which corresponds to Option (D).