An angle is greater than its complementary angle by `30^@`. Find the measure of the angle.

Options

A. `30^@`
B. `90^@`
C. `60^@`
D. `120^@`

Answer

To find the measure of the angle that is greater than its complementary angle by \(30^\circ\), we can follow these steps:

### Step 1: Define the angles
Let \(x\) be the angle we want to find, and let \(y\) be its complementary angle. By definition, complementary angles add up to \(90^\circ\). Therefore, we can write:
\[ x + y = 90^\circ \]

### Step 2: Express the relationship between the angles
According to the problem, the angle \(x\) is greater than its complementary angle \(y\) by \(30^\circ\). This gives us the second equation:
\[ x = y + 30^\circ \]

### Step 3: Substitute the second equation into the first
Now, we can substitute the expression for \(x\) from the second equation into the first equation:
\[ (y + 30^\circ) + y = 90^\circ \]

### Step 4: Simplify the equation
Combine like terms:
\[ 2y + 30^\circ = 90^\circ \]

### Step 5: Solve for \(y\)
Subtract \(30^\circ\) from both sides:
\[ 2y = 90^\circ - 30^\circ \]
\[ 2y = 60^\circ \]
Now, divide both sides by \(2\):
\[ y = 30^\circ \]

### Step 6: Find the value of \(x\)
Now that we have \(y\), we can find \(x\) using the second equation:
\[ x = y + 30^\circ \]
Substituting \(y = 30^\circ\):
\[ x = 30^\circ + 30^\circ = 60^\circ \]

### Conclusion
The measure of the angle \(x\) is \(60^\circ\).