The complement of `65^(@)50'` is :

Options

A. `24^(@)50'`
B. `24^(@)10'`
C. `14^(@)50'`
D. `34^(@)10'`

Answer

To find the complement of the angle \( 65^\circ 50' \), we follow these steps:

### Step 1: Understand the concept of complementary angles
Complementary angles are two angles that add up to \( 90^\circ \). If we have one angle \( A \), its complement \( B \) can be found using the formula:
\[ A + B = 90^\circ \]

### Step 2: Set up the equation
Let the complement of \( 65^\circ 50' \) be \( x \). Thus, we can write:
\[ 65^\circ 50' + x = 90^\circ \]

### Step 3: Convert \( 90^\circ \) into degrees and minutes
Since \( 90^\circ \) can be expressed in degrees and minutes, we note that:
\[ 90^\circ = 89^\circ 60' \]
This is because \( 60' \) represents 1 degree.

### Step 4: Rearrange the equation
Now we can rearrange our equation to solve for \( x \):
\[ x = 90^\circ - 65^\circ 50' \]
Substituting \( 90^\circ \) with \( 89^\circ 60' \):
\[ x = 89^\circ 60' - 65^\circ 50' \]

### Step 5: Perform the subtraction
Now we perform the subtraction:
1. Subtract the minutes:
\[ 60' - 50' = 10' \]
2. Subtract the degrees:
\[ 89^\circ - 65^\circ = 24^\circ \]

### Step 6: Combine the results
Combining the results from the subtraction gives us:
\[ x = 24^\circ 10' \]

### Final Answer
Thus, the complement of \( 65^\circ 50' \) is:
\[ 24^\circ 10' \]

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