The side BC of `DeltaABC` is produced to D. If `angleACD=114^(@)` and `angleABC=((1)/(2)) angleBAC`, what is the value (in degrees) of `angleBAC`?

Options

A. 36
B. 48
C. 76
D. 84

Answer

To solve the problem, we need to find the value of angle BAC in triangle ABC, given that angle ACD = 114° and angle ABC = (1/2) angle BAC.

### Step-by-Step Solution:

1. **Identify the Angles**:
- Let angle BAC = x.
- Then angle ABC = (1/2)x.

2. **Use the Straight Line Property**:
- Since angle ACD = 114°, we can find angle ACB using the straight line property:
\[ \text{Angle ACD} + \text{Angle ACB} = 180° \]
- Therefore,
\[ 114° + \text{Angle ACB} = 180° \]
- Rearranging gives:
\[ \text{Angle ACB} = 180° - 114° = 66° \]

3. **Apply the Triangle Sum Property**:
- The sum of the angles in triangle ABC is 180°:
\[ \text{Angle ABC} + \text{Angle BAC} + \text{Angle ACB} = 180° \]
- Substituting the known values:
\[ \left(\frac{1}{2}x\right) + x + 66° = 180° \]

4. **Combine Like Terms**:
- Combine the terms involving x:
\[ \frac{1}{2}x + x = \frac{3}{2}x \]
- Thus, we have:
\[ \frac{3}{2}x + 66° = 180° \]

5. **Isolate x**:
- Subtract 66° from both sides:
\[ \frac{3}{2}x = 180° - 66° = 114° \]

6. **Solve for x**:
- Multiply both sides by \(\frac{2}{3}\):
\[ x = 114° \times \frac{2}{3} = 76° \]

7. **Conclusion**:
- Therefore, the value of angle BAC is:
\[ \boxed{76°} \]