Parallel Lines Intersection Geometry

A parallelogram is formed when two distinct lines from one set of parallel lines intersect with two distinct lines from another set of parallel lines.

We have two sets of parallel lines:

• Set 1: 4 parallel lines
• Set 2: 5 parallel lines

To form a parallelogram, we need to choose 2 lines from Set 1 and 2 lines from Set 2.

Calculating the Number of Parallelograms

The number of ways to choose 2 lines from a set of 'n' lines is given by the combination formula $\binom{n}{2}$, which is calculated as:

$ \binom{n}{2} = \frac{n!}{2!(n-2)!} = \frac{n \times (n-1)}{2} $

Step 1: Combinations from the first set

Number of ways to choose 2 lines from the 4 parallel lines in Set 1:

$ \binom{4}{2} = \frac{4 \times (4-1)}{2} = \frac{4 \times 3}{2} = \frac{12}{2} = 6 $

Step 2: Combinations from the second set

Number of ways to choose 2 lines from the 5 parallel lines in Set 2:

$ \binom{5}{2} = \frac{5 \times (5-1)}{2} = \frac{5 \times 4}{2} = \frac{20}{2} = 10 $

Step 3: Total Parallelograms

To find the total number of parallelograms, multiply the number of ways to choose lines from each set:

Total Parallelograms = (Ways to choose from Set 1) $\times$ (Ways to choose from Set 2)

Total Parallelograms = $ \binom{4}{2} \times \binom{5}{2} = 6 \times 10 = 60 $

Result

Therefore, 60 parallelograms are formed by the intersection of these two sets of parallel lines.