To solve for the size of angle $\angle HAC$ in the given regular octagon $ABCDEFGH$, we need to break down the geometric relationships.

Steps:

1. Properties of a Regular Octagon:

   • A regular octagon has 8 equal sides and 8 equal angles.
   • The sum of the interior angles of any polygon is given by the formula: $\text{Sum of interior angles} = 180^\circ \times (n - 2)$ where $n$ is the number of sides. For an octagon, $n = 8$, so: $\text{Sum of interior angles} = 180^\circ \times (8 - 2) = 1080^\circ$ Each interior angle in a regular octagon is: $\frac{1080^\circ}{8} = 135^\circ$

2. Consider Triangle $HAC$:

   • Since we are asked to find $\angle HAC$, we note that lines $HA$ and $AC$ form two diagonals within the octagon.
   • To solve this, note that $\triangle HAC$ spans three sides of the octagon (from $H$ to $A$ to $C$).
   • The key is recognizing that the external angle at vertex $A$ (the angle formed outside the octagon) is $45^\circ$, since the external angles of any regular polygon sum to $360^\circ$, and for an octagon, each external angle is: $\frac{360^\circ}{8} = 45^\circ$

3. Angle $\angle HAC$:

   • The diagonals in a regular octagon subtend certain predictable angles. $\angle HAC$ spans across two vertices (from $H$ to $A$ to $C$). Since each exterior angle is $45^\circ$ and this forms a triangle across three vertices of the octagon, the angle $\angle HAC$ is equivalent to twice the exterior angle: $\angle HAC = 2 \times 45^\circ = 90^\circ$

Thus, the size of angle $\angle HAC$ is 90 degrees.

Would you like a more detailed breakdown of any steps or more geometric explanations?

Here are five follow-up questions to expand understanding:

1. How do you calculate the sum of interior angles for any polygon?
2. What are the exterior angles of any regular polygon, and how do they relate to the interior angles?
3. How are diagonals in regular polygons related to angles formed within the polygon?
4. Can you generalize this method to find angles in regular polygons with more or fewer sides?
5. How does symmetry help in determining the properties of regular polygons?

Tip: When working with regular polygons, symmetry is your best tool for simplifying angle relationships!