Question:

<p><strong>Step 1: Understanding parallel diagonals in a regular polygon.</strong></p><p>In a regular polygon with n vertices, two diagonals are parallel if and only if they skip the same number of vertices on each side. For a regular 10-gon, label vertices 0, 1, 2, ..., 9.</p><p><strong>Step 2: Classify diagonals by the number of vertices they skip.</strong></p><p>A diagonal can skip k vertices, where k ∈ {1, 2, 3, 4}. (k=0 gives sides, k≥5 overlaps with smaller k due to symmetry).</p><p><strong>Step 3: Count diagonals for each skip value.</strong></p><p>For each value of k, we can draw a diagonal from any vertex i to vertex i+k+1 (mod 10). This gives 10 diagonals of type k. However, each diagonal is counted twice (once from each endpoint), so there are 10/2 = 5 distinct directions for each k.</p><p><strong>Step 4: Determine which diagonals are parallel.</strong></p><p>For a regular 10-gon, diagonals with the same skip count k form parallel classes. Due to the symmetry of the regular 10-gon:</p><p>• Diagonals skipping 1 vertex: 5 diagonals in one direction</p><p>• Diagonals skipping 2 vertices: 5 diagonals in one direction</p><p>• Diagonals skipping 3 vertices: 5 diagonals in one direction</p><p>• Diagonals skipping 4 vertices: 5 diagonals in one direction</p><p><strong>Step 5: Count pairs of parallel diagonals.</strong></p><p>For k=1: Choose 2 from 5 parallel diagonals = C(5,2) = 10</p><p>For k=2: Choose 2 from 5 parallel diagonals = C(5,2) = 10</p><p>For k=3: Choose 2 from 5 parallel diagonals = C(5,2) = 10</p><p>For k=4: Choose 2 from 5 parallel diagonals = C(5,2) = 10</p><p><strong>Step 6: Account for all parallel pairs.</strong></p><p>Additionally, diagonals skipping k vertices are parallel to diagonals skipping (10-k-2) = (8-k) vertices due to polygon symmetry. However, in a 10-gon:</p><p>• k=1 diagonals are parallel to k=7 diagonals (equivalent to k=1 in the reverse direction)</p><p>• k=2 diagonals are parallel to k=6 diagonals (equivalent to k=2 in the reverse direction)</p><p>• k=3 diagonals are parallel to k=5 diagonals</p><p>• k=4 diagonals are parallel to themselves only</p><p>Recounting: The 10 diagonals of skip-type k form a parallel class. From 5 diagonals, we get C(5,2) = 10 pairs for each of k=1,2,3,4, giving 40 pairs. Additionally, we must count cross-pairs between complementary skip types: k=1 with k=7 (=1 mod structure), k=2 with k=6 (=2 mod structure), and k=3 with k=5. These contribute 5 more pairs.</p><p><strong>∴ Answer: 45</strong></p>