In the diagram, AB, BC and CD are three sides of a regular polygon P - Edexcel - GCSE Maths - Question 5 - 2017 - Paper 3

The exterior angle can be found using the formula:

$\text{Exterior Angle} = 180^{\circ} - \text{Interior Angle}$

For our dodecagon:

$\text{Exterior Angle} = 180^{\circ} - 150^{\circ} = 30^{\circ}.$

This means that each exterior angle of the dodecagon is (30^{\circ}).

For polygon P, which consists of segments AB, BC, and CD, we note that the angles at B and C form exterior angles. Since polygon P is adjacent to the dodecagon, it shares the exterior angles. Thus, if the exterior angle of polygon P is also (30^{\circ}), we can determine the number of sides of polygon P:

Using the formula for the exterior angles:

$\text{Exterior Angle} = \frac{360}{m}$

where ( m ) is the number of sides. Setting (30^{\circ} = \frac{360}{m}$$ gives:

$m = \frac{360}{30} = 12.$

This is the number of sides for the polygon adjacent to the square shape.

Given that polygon P has three sides coinciding with the sides of two squares and making up a total of 6 angles (two each from the squares), we can deduce that polygon P, after including these angles, must be a regular hexagon. In conclusion, we have shown, through working, that polygon P is a hexagon.