In `/_\\ `ABC, side BC has been produced to D. If `/_`ACD= `132^(@)` and `/_`A = `54^(@)`, then `/_`B = ?

Options

A. `48^(@)`.
B. `78^(@)`
C. `68^(@)`
D. `58^(@)`

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the given information
We have triangle ABC, with side BC extended to point D. We know:
- Angle ACD = 132 degrees (this is an exterior angle)
- Angle A = 54 degrees (this is an interior angle)

### Step 2: Use the property of exterior angles
According to the property of triangles, the exterior angle (in this case, angle ACD) is equal to the sum of the two opposite interior angles (angles A and B). This can be expressed as:
\[ \text{Angle ACD} = \text{Angle A} + \text{Angle B} \]

### Step 3: Set up the equation
Substituting the known values into the equation gives us:
\[ 132^\circ = 54^\circ + \text{Angle B} \]

### Step 4: Solve for Angle B
To find Angle B, we will rearrange the equation:
\[ \text{Angle B} = 132^\circ - 54^\circ \]

Now, we perform the subtraction:
\[ \text{Angle B} = 78^\circ \]

### Final Answer
Thus, the measure of angle B is:
\[ \text{Angle B} = 78^\circ \]

---