If the angles `(2a-30^(@)) and (b+60^(@))` make a linear pair, find the values of a and b when `a-b=30^(@)`

Answer

To solve the problem, we need to find the values of \( a \) and \( b \) given that the angles \( (2a - 30^\circ) \) and \( (b + 60^\circ) \) form a linear pair and that \( a - b = 30^\circ \).

### Step-by-Step Solution:

1. **Understanding Linear Pair**:
Since the angles \( (2a - 30^\circ) \) and \( (b + 60^\circ) \) form a linear pair, we know that their sum is equal to \( 180^\circ \).
\[ (2a - 30^\circ) + (b + 60^\circ) = 180^\circ \]

2. **Setting Up the Equation**:
Simplifying the above equation:
\[ 2a - 30^\circ + b + 60^\circ = 180^\circ \]
Combine like terms:
\[ 2a + b + 30^\circ = 180^\circ \]
Now, subtract \( 30^\circ \) from both sides:
\[ 2a + b = 150^\circ \quad \text{(Equation 1)} \]

3. **Using the Second Condition**:
We are also given that \( a - b = 30^\circ \).
\[ a - b = 30^\circ \quad \text{(Equation 2)} \]

4. **Solving the Equations**:
Now we have two equations:
- Equation 1: \( 2a + b = 150^\circ \)
- Equation 2: \( a - b = 30^\circ \)

We can solve these equations simultaneously. Let's add Equation 1 and Equation 2:
\[ (2a + b) + (a - b) = 150^\circ + 30^\circ \]
This simplifies to:
\[ 3a = 180^\circ \]
Now, divide both sides by 3:
\[ a = 60^\circ \]

5. **Finding \( b \)**:
Now that we have \( a \), we can substitute it back into Equation 2 to find \( b \):
\[ 60^\circ - b = 30^\circ \]
Rearranging gives:
\[ b = 60^\circ - 30^\circ = 30^\circ \]

### Final Values:
Thus, the values are:
\[ a = 60^\circ, \quad b = 30^\circ \]