Let x be the angle which is equal to its complement and y be the angle which is equal to its supplement. Then, 2x + 3y is equal to

Options

A. `135^(@)`
B. `270^(@)`
C. `360^(@)`
D. `180^(@)`

Answer

To solve the problem, we need to find the values of \( x \) and \( y \) based on the definitions of complementary and supplementary angles.

### Step 1: Define the angles
Let \( x \) be the angle which is equal to its complement. By definition, the complement of an angle \( x \) is \( 90^\circ - x \).

### Step 2: Set up the equation for \( x \)
Since \( x \) is equal to its complement, we can write:
\[ x = 90^\circ - x \]

### Step 3: Solve for \( x \)
Now, we can solve the equation:
\[ x + x = 90^\circ \]
\[ 2x = 90^\circ \]
\[ x = \frac{90^\circ}{2} = 45^\circ \]

### Step 4: Define the angle \( y \)
Let \( y \) be the angle which is equal to its supplement. By definition, the supplement of an angle \( y \) is \( 180^\circ - y \).

### Step 5: Set up the equation for \( y \)
Since \( y \) is equal to its supplement, we can write:
\[ y = 180^\circ - y \]

### Step 6: Solve for \( y \)
Now, we can solve the equation:
\[ y + y = 180^\circ \]
\[ 2y = 180^\circ \]
\[ y = \frac{180^\circ}{2} = 90^\circ \]

### Step 7: Calculate \( 2x + 3y \)
Now that we have \( x = 45^\circ \) and \( y = 90^\circ \), we can find \( 2x + 3y \):
\[ 2x + 3y = 2(45^\circ) + 3(90^\circ) \]
\[ = 90^\circ + 270^\circ \]
\[ = 360^\circ \]

### Final Answer
Thus, \( 2x + 3y = 360^\circ \).

---