1. **State the problem:** We are given two sides AB and BC of a regular polygon with $n$ sides, and two angles at vertex B: $148^\circ$ and $50^\circ$. We need to find the number of sides $n$ of the polygon. 2. **Understand the polygon and angles:** In a regular polygon, all interior angles are equal, and all sides are equal. The exterior angle $E$ of a regular polygon with $n$ sides is given by the formula: $$E = \frac{360^\circ}{n}$$ 3. **Analyze the given angles:** The angle between AB and the vertical line through B is $148^\circ$, and the angle between BC and the extension of AB is $50^\circ$. Since AB and BC are consecutive sides, the angle between them at vertex B is the interior angle of the polygon. 4. **Calculate the interior angle:** The interior angle $I$ at vertex B is the sum of the two given angles: $$I = 148^\circ + 50^\circ = 198^\circ$$ 5. **Relate interior and exterior angles:** The interior and exterior angles are supplementary: $$I + E = 180^\circ$$ So, $$E = 180^\circ - I = 180^\circ - 198^\circ = -18^\circ$$ Since an exterior angle cannot be negative, we must reconsider the interpretation. The $148^\circ$ angle is likely the reflex angle outside the polygon, so the interior angle is actually: $$I = 360^\circ - 148^\circ - 50^\circ = 162^\circ$$ 6. **Calculate the exterior angle again:** $$E = 180^\circ - I = 180^\circ - 162^\circ = 18^\circ$$ 7. **Find the number of sides $n$:** $$n = \frac{360^\circ}{E} = \frac{360^\circ}{18^\circ} = 20$$ **Final answer:** $$\boxed{20}$$