In the given figure, x:y = 2:3 and `/_ACD = 120^(@)`. Find the values of x y and z.

Answer

To solve the problem, we need to find the values of angles x, y, and z given that the ratio x:y = 2:3 and angle ACD = 120 degrees.

### Step-by-Step Solution:

1. **Understanding the Ratio**:
Given the ratio x:y = 2:3, we can express y in terms of x.
Let x = 2k and y = 3k for some constant k.

2. **Using the Exterior Angle Theorem**:
According to the Exterior Angle Theorem, the exterior angle (angle ACD) is equal to the sum of the two opposite interior angles (x and y).
Therefore, we can write the equation:
\[ x + y = 120^\circ \]

3. **Substituting the Values**:
Substitute the expressions for x and y into the equation:
\[ 2k + 3k = 120^\circ \]
This simplifies to:
\[ 5k = 120^\circ \]

4. **Solving for k**:
Divide both sides by 5 to find k:
\[ k = \frac{120^\circ}{5} = 24^\circ \]

5. **Finding x and y**:
Now substitute k back into the expressions for x and y:
\[ x = 2k = 2 \times 24^\circ = 48^\circ \]
\[ y = 3k = 3 \times 24^\circ = 72^\circ \]

6. **Finding z**:
In a triangle, the sum of the angles is always 180 degrees. Therefore, we can find z as follows:
\[ z = 180^\circ - (x + y) = 180^\circ - (48^\circ + 72^\circ) = 180^\circ - 120^\circ = 60^\circ \]

### Final Values:
- \( x = 48^\circ \)
- \( y = 72^\circ \)
- \( z = 60^\circ \)