If `alpha` lies in ll quadrant, `beta` lies in III quadrant and `tan (alpha + beta) gt 0`, then `(alpha + beta)` lies in... quadrants.

Answer

To solve the problem, we need to analyze the given information about the angles \( \alpha \) and \( \beta \) and determine the quadrant in which \( \alpha + \beta \) lies.

### Step 1: Identify the quadrants of \( \alpha \) and \( \beta \)
- \( \alpha \) lies in the second quadrant.
- \( \beta \) lies in the third quadrant.

### Step 2: Determine the general angles for \( \alpha \) and \( \beta \)
- In the second quadrant, angles can be expressed as:
\[ \alpha = \pi - \theta \quad (\text{where } 0 < \theta < \pi) \]
- In the third quadrant, angles can be expressed as:
\[ \beta = \pi + \phi \quad (\text{where } 0 < \phi < \pi) \]

### Step 3: Calculate \( \alpha + \beta \)
Now, we can find \( \alpha + \beta \):
\[ \alpha + \beta = (\pi - \theta) + (\pi + \phi) = 2\pi + \phi - \theta \]

### Step 4: Analyze the expression \( 2\pi + \phi - \theta \)
Since \( \phi \) and \( \theta \) are both positive angles (as they are in their respective quadrants), the expression \( \phi - \theta \) can be either positive, negative, or zero. However, since \( 2\pi \) is added, the overall angle \( \alpha + \beta \) will be greater than \( 2\pi \).

### Step 5: Determine the quadrant of \( \alpha + \beta \)
- Angles greater than \( 2\pi \) will wrap around back into the standard position.
- Since \( \alpha + \beta = 2\pi + (\phi - \theta) \), we need to consider the value of \( \phi - \theta \):
- If \( \phi - \theta \) is positive, \( \alpha + \beta \) is in the first quadrant.
- If \( \phi - \theta \) is negative, \( \alpha + \beta \) is in the fourth quadrant.
- If \( \phi - \theta = 0\), \( \alpha + \beta \) is exactly \( 2\pi \) (which is equivalent to \( 0 \) radians).

However, since we are given that \( \tan(\alpha + \beta) > 0 \), this implies that \( \alpha + \beta \) must be in a quadrant where the tangent function is positive. The tangent function is positive in the first and third quadrants.

### Conclusion
Since \( \alpha \) is in the second quadrant and \( \beta \) is in the third quadrant, and given that \( \tan(\alpha + \beta) > 0 \), we conclude that \( \alpha + \beta \) lies in the **first quadrant**.