The number of segments formed by joining $n$ distinct points is given by the combination formula $\binom{n}{2} = \frac{n(n-1)}{2}$.

Part 1: Segments formed by joining six collinear points. For $n=6$ collinear points, the number of segments is $\binom{6}{2} = \frac{6 \times (6-1)}{2} = \frac{6 \times 5}{2} = \frac{30}{2} = 15$.

Part 2: Segments formed by joining five non-collinear points. For $n=5$ non-collinear points, the number of segments is $\binom{5}{2} = \frac{5 \times (5-1)}{2} = \frac{5 \times 4}{2} = \frac{20}{2} = 10$.

To find how many more segments are formed, we subtract the second number from the first: Difference $= 15 - 10 = 5$. Therefore, $5$ more segments are formed.