Question

Two complementary angles are such that two times the measure of one is equal to three times the measure of the other. The measure of the smaller angle is :

• A. $45^\circ$
• B. $30^\circ$
• C. $36^\circ$
• D. None of these

Hint:

Remember that "complementary" means angles sum to $90^\circ$, while "supplementary" means angles sum to $180^\circ$. Always set up two equations from the given information to solve for two unknown angles.

Answer 1

The Correct Option is C

Step 1: Define complementary angles and set up equations.
Complementary angles are two angles whose sum is $90^\circ$.
Let the two angles be $x$ and $y$.
From the definition of complementary angles:
$x + y = 90^\circ$ (Equation 1) From the problem statement, "two times the measure of one is equal to three times the measure of the other":
$2x = 3y$ (Equation 2) Step 2: Solve the system of equations.
From Equation 2, express one variable in terms of the other.
$x = \frac{3}{2}y$ Substitute this expression for $x$ into Equation 1:
$\frac{3}{2}y + y = 90^\circ$
Multiply by 2 to eliminate the fraction:
$3y + 2y = 180^\circ$
$5y = 180^\circ$
$y = \frac{180^\circ}{5}$
$y = 36^\circ$
Now, find $x$ using Equation 1:
$x + 36^\circ = 90^\circ$
$x = 90^\circ - 36^\circ$
$x = 54^\circ$
Step 3: Identify the smaller angle.
The two angles are $54^\circ$ and $36^\circ$.
The smaller angle is $36^\circ$.
Step 4: Compare with the given options.
The calculated smaller angle is $36^\circ$, which matches option (3). (3) 36\textdegree