The side BC of a `DeltaABC` is extended to D. If `angleACD = 120^(@)` and `angleABC =(1)/(2)angleCAB`, then the value of `angleABC` is

Options

A. `80^(@)`
B. `40^(@)`
C. `60^(@)`
D. `20^(@)`

Answer

To solve the problem, we need to find the value of angle ABC given that angle ACD = 120° and angle ABC = (1/2) * angle CAB. Let's denote angle CAB as \( x \) and angle ABC as \( y \).

### Step-by-step Solution:

1. **Identify the angles:**
- Let \( \angle ABC = y \)
- Let \( \angle CAB = x \)
- Given \( \angle ACD = 120° \)

2. **Use the relationship between angles:**
- We know that \( \angle ABC = \frac{1}{2} \angle CAB \), which gives us:
\[ y = \frac{1}{2} x \]

3. **Apply the angle sum property in triangle ABC:**
- In triangle ABC, the sum of the angles is 180°:
\[ x + y + \angle ACB = 180° \]
- We also know that \( \angle ACB + \angle ACD = 180° \) because they are supplementary angles. Therefore:
\[ \angle ACB + 120° = 180° \]
- This implies:
\[ \angle ACB = 180° - 120° = 60° \]

4. **Substitute the known values into the triangle angle sum equation:**
- Now substituting \( \angle ACB = 60° \) into the angle sum equation:
\[ x + y + 60° = 180° \]

5. **Simplify the equation:**
- Rearranging gives:
\[ x + y = 180° - 60° = 120° \]

6. **Substitute \( y \) in terms of \( x \):**
- From step 2, we have \( y = \frac{1}{2} x \). Substitute this into the equation:
\[ x + \frac{1}{2} x = 120° \]

7. **Combine like terms:**
- This simplifies to:
\[ \frac{3}{2} x = 120° \]

8. **Solve for \( x \):**
- Multiply both sides by \( \frac{2}{3} \):
\[ x = 120° \cdot \frac{2}{3} = 80° \]

9. **Find \( y \):**
- Now substitute \( x \) back to find \( y \):
\[ y = \frac{1}{2} x = \frac{1}{2} \cdot 80° = 40° \]

### Final Answer:
Thus, the value of \( \angle ABC \) is \( 40° \).