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 larger angle is :

Options

A. `72^(@)`
B. `108^(@)`
C. `36^(@)`
D. `54^(@)`

Answer

To solve the problem of finding the measure of the larger angle among two complementary angles where two times the measure of one angle is equal to three times the measure of the other, we can follow these steps:

### Step-by-Step Solution:

1. **Define the angles**: Let the two angles be \( x \) and \( y \).

2. **Set up the complementary angle equation**: Since the angles are complementary, we know that:
\[ x + y = 90^\circ \quad \text{(Equation 1)} \]

3. **Set up the relationship between the angles**: According to the problem, two times the measure of one angle is equal to three times the measure of the other. We can express this as:
\[ 2x = 3y \quad \text{(Equation 2)} \]

4. **Substitute Equation 1 into Equation 2**: From Equation 1, we can express \( y \) in terms of \( x \):
\[ y = 90 - x \]
Now substitute this into Equation 2:
\[ 2x = 3(90 - x) \]

5. **Solve for \( x \)**: Expand and rearrange the equation:
\[ 2x = 270 - 3x \]
Add \( 3x \) to both sides:
\[ 2x + 3x = 270 \]
\[ 5x = 270 \]
Divide both sides by 5:
\[ x = \frac{270}{5} = 54^\circ \]

6. **Find \( y \)**: Now substitute \( x \) back into Equation 1 to find \( y \):
\[ y = 90 - x = 90 - 54 = 36^\circ \]

7. **Identify the larger angle**: The two angles are \( 54^\circ \) and \( 36^\circ \). Therefore, the measure of the larger angle is:
\[ \text{Larger angle} = 54^\circ \]

### Final Answer:
The measure of the larger angle is \( 54^\circ \).