Calculate the angles x,y and z if
` (x)/(3) = (y)/(4) =(z)/(5)`
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Answer

To solve the problem of finding the angles x, y, and z given that \( \frac{x}{3} = \frac{y}{4} = \frac{z}{5} \), we can follow these steps:

### Step-by-Step Solution

1. **Set a Common Variable**:
Let \( k \) be the common ratio. Thus, we can express x, y, and z in terms of k:
\[ x = 3k, \quad y = 4k, \quad z = 5k \]

2. **Use the Property of Cyclic Quadrilaterals**:
In a cyclic quadrilateral, the sum of opposite angles is equal to 180 degrees. Therefore, we can write:
\[ \text{Angle ABC} + \text{Angle ADC} = 180^\circ \]
Here, Angle ABC is \( x + y \) and Angle ADC is \( x + z \).

3. **Set Up the Equation**:
Substitute the expressions for x, y, and z into the equation:
\[ (x + y) + (x + z) = 180^\circ \]
This simplifies to:
\[ (3k + 4k) + (3k + 5k) = 180^\circ \]
Which further simplifies to:
\[ 2(3k + 4k) = 180^\circ \]
or
\[ 2(6k) = 180^\circ \]

4. **Solve for k**:
Divide both sides by 2:
\[ 6k = 90^\circ \]
Now, divide by 6:
\[ k = 15^\circ \]

5. **Calculate x, y, and z**:
Substitute \( k \) back into the expressions for x, y, and z:
\[ x = 3k = 3 \times 15^\circ = 45^\circ \]
\[ y = 4k = 4 \times 15^\circ = 60^\circ \]
\[ z = 5k = 5 \times 15^\circ = 75^\circ \]

### Final Values:
- \( x = 45^\circ \)
- \( y = 60^\circ \)
- \( z = 75^\circ \)