ABC is a triangle and the sides AB, BC and CA are produced to E, F and G respectively. If `/_CBE = /_ACF = 130^@` then the value of `/_GAB` is

Options

A. `100^@`
B. `130^@`
C. `80^@`
D. `90^@`

Answer

To solve the problem step by step, we will analyze the triangle ABC and the angles given.

### Step 1: Understand the Triangle and Given Angles
We have triangle ABC with sides extended to points E, F, and G. We know:
- Angle CBE = 130°
- Angle ACF = 130°

### Step 2: Identify Angles X and Y
Let:
- Angle GAB = X
- Angle ABC = Y

### Step 3: Apply Linear Pair on Angle CBE
Since angle CBE and angle GAB (X) are on a straight line:
\[ X + \text{Angle CBE} = 180° \]
Substituting the value of Angle CBE:
\[ X + 130° = 180° \]
Thus,
\[ X = 180° - 130° = 50° \]

### Step 4: Apply Linear Pair on Angle ACF
Similarly, for angle ACF:
\[ Y + \text{Angle ACF} = 180° \]
Substituting the value of Angle ACF:
\[ Y + 130° = 180° \]
Thus,
\[ Y = 180° - 130° = 50° \]

### Step 5: Apply Angle Sum Property in Triangle ABC
The sum of angles in triangle ABC is:
\[ X + Y + \text{Angle BAC} = 180° \]
Substituting the values of X and Y:
\[ 50° + 50° + \text{Angle BAC} = 180° \]
This simplifies to:
\[ 100° + \text{Angle BAC} = 180° \]
Thus,
\[ \text{Angle BAC} = 180° - 100° = 80° \]

### Step 6: Apply Linear Pair on Angle CAG
Now, we apply the linear pair on angle CAG:
\[ \text{Angle BAC} + \text{Angle GAB} = 180° \]
Substituting the value of Angle BAC:
\[ 80° + \text{Angle GAB} = 180° \]
Thus,
\[ \text{Angle GAB} = 180° - 80° = 100° \]

### Conclusion
The value of angle GAB is **100°**.

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