The problem presents a comparison of two quantities regarding the diagonals of a regular octagon:

• Quantity A: The number of diagonals of the octagon that are parallel to at least one side of the octagon.
• Quantity B: The number of diagonals of the octagon that are not parallel to any side of the octagon.

Step 1: Total number of diagonals in an octagon

The formula for calculating the number of diagonals in a polygon with $n$ sides is:

$\text{Number of diagonals} = \frac{n(n-3)}{2}$

For a regular octagon ($n = 8$):

$\text{Number of diagonals} = \frac{8(8-3)}{2} = \frac{8 \times 5}{2} = 20$

So, there are 20 diagonals in a regular octagon.

Step 2: Understanding diagonal properties

• Parallel diagonals: Diagonals that are parallel to one of the sides of the octagon.
• Non-parallel diagonals: Diagonals that are not parallel to any side of the octagon.

A regular octagon has symmetry, and diagonals of different lengths. Some diagonals may be parallel to the sides, but not all.

Step 3: Comparing quantities

It turns out that only 4 diagonals in a regular octagon are parallel to its sides. These diagonals are the ones that skip exactly two vertices.

This means that:

• Quantity A (diagonals parallel to at least one side) = 4 diagonals.
• Quantity B (diagonals not parallel to any side) = Total diagonals (20) minus diagonals that are parallel (4), which equals 16 diagonals.

Final comparison:

Since Quantity A (4) is clearly less than Quantity B (16), the correct relationship is:

$\text{Quantity A} < \text{Quantity B}$

Would you like further details or have any questions?

Here are 5 related questions to consider:

1. How can we calculate the number of diagonals for any polygon?
2. What are the different types of diagonals in polygons with more sides?
3. How does symmetry influence the properties of diagonals in polygons?
4. How does the number of diagonals change as the number of sides increases?
5. Why do certain diagonals become parallel to sides in regular polygons?

Tip: In polygons with many sides, symmetry plays a crucial role in determining the properties of diagonals, which can help simplify many problems.