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The problem involves a regular 8-sided polygon (octagon) and a regular 10-sided polygon (decagon) sharing a common side, and we are asked to calculate the size of the angle marked \( b \) at the intersection of these polygons.

### Step 1: Calculate the interior angles of the polygons
For any regular polygon with \( n \) sides, the formula for the interior angle \( \theta \) is given by:

\[
\theta = \frac{(n-2) \times 180^\circ}{n}
\]

- **For the octagon (8-sided polygon):**

\[
\theta_{\text{octagon}} = \frac{(8-2) \times 180^\circ}{8} = \frac{6 \times 180^\circ}{8} = 135^\circ
\]

- **For the decagon (10-sided polygon):**

\[
\theta_{\text{decagon}} = \frac{(10-2) \times 180^\circ}{10} = \frac{8 \times 180^\circ}{10} = 144^\circ
\]

### Step 2: Calculate the exterior angles of the polygons
The exterior angle \( \phi \) of a polygon is related to its interior angle by:

\[
\phi = 180^\circ - \theta
\]

- **For the octagon:**

\[
\phi_{\text{octagon}} = 180^\circ - 135^\circ = 45^\circ
\]

- **For the decagon:**

\[
\phi_{\text{decagon}} = 180^\circ - 144^\circ = 36^\circ
\]

### Step 3: Angle at point \( b \)
At the vertex where the two polygons meet, the exterior angles combine to form a straight line. The sum of these angles must be 180° since they lie along a straight line. Hence, the angle \( b \) is:

\[
b = 180^\circ - (\phi_{\text{octagon}} + \phi_{\text{decagon}})
\]

\[
b = 180^\circ - (45^\circ + 36^\circ) = 180^\circ - 81^\circ = 99^\circ
\]

### Final Answer:
The size of the angle marked \( b \) is \( \boxed{99^\circ} \).

Let me know if you'd like more details, or if you have any questions!

Here are 5 questions that expand on this concept:
1. How would the calculation change if one polygon were a pentagon instead of a decagon?
2. What is the general formula for the exterior angle of a regular polygon?
3. Can the angle \( b \) ever be less than \( 90^\circ \) for any regular polygons?
4. How can you determine the sum of interior angles of a polygon without calculating each individual angle?
5. What happens to the angle \( b \) as the number of sides of one of the polygons increases?

**Tip**: Always remember that the sum of exterior angles of any polygon is always \( 360^\circ \), no matter how many sides it has!

A regular 8-sided polygon and a regular 10-sided polygon share a single side. Calculate the size of the angle marked b.

Learn how to calculate the size of the angle formed when a regular 8-sided polygon (octagon) and a regular 10-sided polygon (decagon) share a side. This problem covers key geometric concepts like polygon interior and exterior angles. Suitable for students in Grades 9-11.

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