The problem shows two congruent regular polygons joined at a vertex, with an exterior angle of 72°. To solve for the number of sides of each polygon, we use the formula for the exterior angle of a regular polygon:

$\text{Exterior angle} = \frac{360^\circ}{n}$

where $n$ is the number of sides.

Step-by-Step Solution:

1. From the diagram, the exterior angle is given as $72^\circ$.
2. Using the exterior angle formula:

$72^\circ = \frac{360^\circ}{n}$

3. Solving for $n$:

$n = \frac{360^\circ}{72^\circ} = 5$

Thus, each polygon is a regular pentagon with 5 sides.

Would you like a further breakdown or have any questions?

Here are five related questions to expand on this concept:

1. What is the sum of the interior angles of a regular pentagon?
2. How do you calculate the interior angle of any regular polygon?
3. If the exterior angle were 60°, how many sides would the polygon have?
4. What is the relationship between exterior and interior angles in regular polygons?
5. How does the number of sides affect the size of the exterior angles of a polygon?

Tip: Remember that the sum of exterior angles for any polygon always adds up to 360°, no matter how many sides!