Example 1 Using the two vertices of a triangle and $n(n \geqslant 1)$ points inside it as vertices, construct all non-overlapping small triangles. Find the number of these small triangles.
Solution: Since each of the $n$ points inside the original triangle is a common vertex of several small triangles, and the sum of the angles at that vertex in these small triangles is $360^{\circ}$, the total sum of the interior angles of all the small triangles is $360^{\circ} \cdot n + 180^{\circ}$. Therefore, the number of these small triangles is
$$
\left(360^{\circ} \cdot n + 180^{\circ}\right) \div 180^{\circ} = 2n + 1.
$$