To solve for the angle $x$ in the image, we need to consider the geometrical properties of the shapes provided (a regular pentagon and two regular hexagons).

Step 1: Angles in a Regular Pentagon and Hexagon

• A regular pentagon has interior angles of $108^\circ$ because the formula for the interior angle of any regular polygon is given by: $\text{Interior angle} = \frac{(n-2) \times 180^\circ}{n}$ where $n$ is the number of sides. For a pentagon, $n = 5$: $\frac{(5-2) \times 180^\circ}{5} = \frac{3 \times 180^\circ}{5} = 108^\circ$

• A regular hexagon has interior angles of $120^\circ$, derived similarly for $n = 6$: $\frac{(6-2) \times 180^\circ}{6} = \frac{4 \times 180^\circ}{6} = 120^\circ$

Step 2: Understanding Angle $x$

The image shows two hexagons and one pentagon adjacent to each other, with angle $x$ between the hexagon and the pentagon. The angle at that vertex is the external angle formed by the intersection of these polygons.

The key observation is that the external angles around a point sum to $360^\circ$. The external angle of a polygon is $180^\circ -$ interior angle. Thus:

• The external angle of the pentagon is: $180^\circ - 108^\circ = 72^\circ$
• The external angle of each hexagon is: $180^\circ - 120^\circ = 60^\circ$

Step 3: Solving for $x$

Since the external angles must sum to $360^\circ$, we can calculate $x$ as the remaining angle at the vertex. Adding up the external angles: $60^\circ (\text{hexagon}) + 60^\circ (\text{other hexagon}) + 72^\circ (\text{pentagon}) + x = 360^\circ$ $60^\circ + 60^\circ + 72^\circ + x = 360^\circ$ $192^\circ + x = 360^\circ$ $x = 360^\circ - 192^\circ = 168^\circ$

Thus, the size of angle $x$ is $168^\circ$.

Would you like more details on any step or have any questions?

Related Questions:

1. How is the formula for the interior angle of a regular polygon derived?
2. What are the external angles of an octagon?
3. If a triangle and a pentagon were placed together, how would you find the angle between them?
4. What would the angle $x$ be if one of the hexagons were replaced with a square?
5. Can you generalize a formula to find $x$ for any combination of polygons at a vertex?

Tip:

For polygons, always remember that the sum of the external angles around a point is always $360^\circ$, regardless of the shapes!