The interior angles of a convex polygon form an arithmetic progression with a common difference of `4^(@)`. Determine the number of sides of the polygon if its largest interior angle is `172^(@)`.

Answer

To solve the problem, we need to find the number of sides of a convex polygon where the interior angles form an arithmetic progression with a common difference of \(4^\circ\) and the largest interior angle is \(172^\circ\).

### Step-by-Step Solution:

1. **Understanding the Problem**:
- The largest interior angle of the polygon is given as \(172^\circ\).
- The angles are in an arithmetic progression (AP) with a common difference of \(4^\circ\).

2. **Finding the Smallest Interior Angle**:
- The second largest angle can be calculated as:
\[ 172^\circ - 4^\circ = 168^\circ \]
- Continuing this pattern, the angles will be:
\[ 172^\circ, 168^\circ, 164^\circ, \ldots \]
- The smallest angle will be \(172^\circ - (n-1) \cdot 4^\circ\), where \(n\) is the number of sides (or angles).

3. **Finding the Corresponding Exterior Angles**:
- The exterior angle corresponding to the largest interior angle is:
\[ 180^\circ - 172^\circ = 8^\circ \]
- The exterior angle corresponding to the second largest angle is:
\[ 180^\circ - 168^\circ = 12^\circ \]
- Thus, the exterior angles also form an AP:
\[ 8^\circ, 12^\circ, 16^\circ, \ldots \]

4. **Sum of Exterior Angles**:
- The sum of all exterior angles of any polygon is \(360^\circ\).
- The exterior angles form an arithmetic series where:
- First term \(a = 8^\circ\)
- Common difference \(d = 4^\circ\)
- Number of terms \(n\)

5. **Using the Sum Formula for AP**:
- The sum \(S_n\) of the first \(n\) terms of an arithmetic progression is given by:
\[ S_n = \frac{n}{2} \cdot (2a + (n-1)d) \]
- Setting this equal to \(360^\circ\):
\[ \frac{n}{2} \cdot (2 \cdot 8 + (n-1) \cdot 4) = 360 \]

6. **Simplifying the Equation**:
- This simplifies to:
\[ \frac{n}{2} \cdot (16 + 4n - 4) = 360 \]
\[ \frac{n}{2} \cdot (4n + 12) = 360 \]
\[ n(4n + 12) = 720 \]
\[ 4n^2 + 12n - 720 = 0 \]

7. **Dividing the Equation**:
- Divide the entire equation by 4:
\[ n^2 + 3n - 180 = 0 \]

8. **Factoring the Quadratic**:
- We can factor this quadratic:
\[ (n - 12)(n + 15) = 0 \]
- Thus, \(n = 12\) or \(n = -15\).

9. **Conclusion**:
- Since \(n\) must be a positive integer (number of sides), we have:
\[ n = 12 \]

### Final Answer:
The number of sides of the polygon is \(12\).