Add `55^(@)49' 31'' and 63^(@) 22' 45''`

Answer

To solve the problem of adding \(55^\circ 49' 31''\) and \(63^\circ 22' 45''\), we will follow these steps:

### Step 1: Align the angles for addition
We will write the two angles one below the other, aligning degrees, minutes, and seconds.

\[ \begin{array}{c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c} & 55^\circ & 49' & 31'' \\ + & 63^\circ & 22' & 45'' \\ \hline \end{array} \]

### Step 2: Add the seconds
We will add the seconds first:

\[ 31'' + 45'' = 76'' \]

### Step 3: Convert seconds to minutes
Since \(76''\) is more than \(60''\), we will convert \(60''\) to \(1'\):

\[ 76'' = 60'' + 16'' \quad \text{(which is } 1' \text{ and } 16'') \]

### Step 4: Add the minutes
Now we add the minutes, including the extra minute from the seconds:

\[ 49' + 22' + 1' = 72' \]

### Step 5: Convert minutes to degrees
Since \(72'\) is more than \(60'\), we will convert \(60'\) to \(1^\circ\):

\[ 72' = 60' + 12' \quad \text{(which is } 1^\circ \text{ and } 12'\text{)} \]

### Step 6: Add the degrees
Finally, we add the degrees, including the extra degree from the minutes:

\[ 55^\circ + 63^\circ + 1^\circ = 119^\circ \]

### Step 7: Combine the results
Now we can combine everything together:

\[ 119^\circ + 12' + 16'' \]

### Final Answer
Thus, the final result of adding \(55^\circ 49' 31''\) and \(63^\circ 22' 45''\) is:

\[ 119^\circ 12' 16'' \]

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