The problem involves calculating the angles $a$ and $b$ based on the properties of the polygons shown in the diagram. Here's how to solve each part:

Step-by-Step Solution

a) Calculate the size of angle $a$:

1. Regular polygons meeting at a point:
   At the vertex where angle $a$ is located, three regular polygons join together. Let the number of sides of each polygon be $n$.
   For a regular polygon with $n$ sides, the interior angle is given by:
   $\text{Interior angle} = \frac{(n-2) \times 180^\circ}{n}$ Since the three interior angles sum to $360^\circ$ at this vertex, we set up the equation: $3 \times \frac{(n-2) \times 180^\circ}{n} = 360^\circ$

2. Solve for $n$:
   Simplify the equation:
   $\frac{3(n-2) \times 180^\circ}{n} = 360^\circ$ Divide through by $180^\circ$:
   $\frac{3(n-2)}{n} = 2$ Multiply through by $n$:
   $3(n-2) = 2n$ Expand and rearrange:
   $3n - 6 = 2n$ $n = 6$ This tells us the polygons are regular hexagons.

3. Angle $a$:
   The interior angle of a regular hexagon is:
   $\frac{(6-2) \times 180^\circ}{6} = 120^\circ$ Since $a$ is the angle between two adjacent edges of the hexagon, it is a supplementary angle to the interior angle:
   $a = 180^\circ - 120^\circ = 60^\circ$

   Answer for (a):
   $a = 60^\circ$

b) Calculate the size of angle $b$:

1. Angle at the center of polygons:
   Four regular hexagons meet at a point to form a quadrilateral, with $b$ as one of its interior angles. The angles at the center of the polygons contribute to $b$.

   The central angle of a regular hexagon is given by:
   $\text{Central angle} = \frac{360^\circ}{n} = \frac{360^\circ}{6} = 60^\circ$ For four polygons, the total angle is:
   $4 \times 60^\circ = 240^\circ$ The remaining angle in the quadrilateral at the center is:
   $b = 360^\circ - 240^\circ = 120^\circ$

   Answer for (b):
   $b = 120^\circ$

Final Answers:

• (a) $a = 60^\circ$
• (b) $b = 120^\circ$

Would you like additional clarification or details?

Here are 5 related questions to explore:

1. How do you calculate the exterior angles of regular polygons?
2. Why do angles around a point always sum to $360^\circ$?
3. How can the sum of interior angles of a polygon be determined?
4. Can you apply the same method to polygons with different numbers of sides?
5. What are the properties of a regular hexagon?

Tip: Always check if the sum of angles matches the geometric constraints (e.g., $360^\circ$ around a point).