<Question>
In a regular polygon, the measure of an interior angle is 6.5 times the measure of an exterior angle. How many sides does the polygon have?
</Question>

<think>
<Plan_1>
Okay, so I have this problem here: In a regular polygon, the measure of an interior angle is 6.5 times the measure of an exterior angle. How many sides does the polygon have? Hmm, let me think. I remember that in regular polygons, the sum of the interior and exterior angles is 180 degrees because they are supplementary. Right? So if the interior angle is 6.5 times the exterior angle, maybe I can set up an equation using that relationship.

Let me denote the measure of the exterior angle as E. Then the interior angle would be 6.5E. Since they are supplementary, E + 6.5E = 180.
That makes sense. So 7.5E = 180. Then E = 180 / 7.5. Let me calculate that. 180 divided by 7.5. Hmm, 7.5 times 24 is 180 because 7.5 times 20 is 150, and 7.5 times 4 is 30, so 150 + 30 = 180.

So the measure of each exterior angle is 24 degrees.

Now, I recall that the sum of all exterior angles of any polygon is 360 degrees. Since this is a regular polygon, all exterior angles are equal, so the number of sides n would be 360 divided by the measure of each exterior angle.
So n = 360 / E. Substituting E = 24, n = 360 / 24. Let me compute that. 24 times 15 is 360 because 24 times 10 is 240, and 24 times 5 is 120, so 240 + 120 = 360. Therefore, n = 15.

Wait, let me verify. If each exterior angle is 24 degrees, then the interior angle is 180 - 24 = 156 degrees. The problem states that the interior angle is 6.5 times the exterior angle.
So 6.5 times 24 should equal 156. Let's check: 6 times 24 is 144, and 0.5 times 24 is 12, so 144 + 12 = 156.

But just to make sure I didn't make any mistakes in the calculations. Let me go through each step again.

<prompt_1.1> Recheck the calculations for the exterior angle and the number of sides. </prompt_1.1>

Alternatively, I can use the formula for the interior angle of a regular polygon: [(n - 2) * 180] / n. Since the interior angle is 156, set that equal to [(n - 2)/n] * 180.

<prompt_1.2> Use an alternative formula to verify the number of sides. </prompt_1.2>

So the polygon has 15 sides.

But wait, just to ensure there's no alternative interpretation of the problem. The problem says the interior angle is 6.5 times the exterior angle.

<prompt_1.3> Consider alternative interpretations of the problem and confirm the solution. </prompt_1.3>

Alternatively, if I use the formula for exterior angles: 360/n = E. So if E = 24, then n = 360/24 = 15. Yep.

Another way to think about it: The ratio of interior to exterior is 6.5:1, which is 13:2.
So parts are 13 parts for interior, 2 parts for exterior. Total parts 15. Each part is 12 degrees (since 2 parts = 24, 1 part =12). So 13 parts would be 156, which is the interior angle.
</Plan_1>

<Execution_1>
<execution_1.1>
Starting with the beginning: interior angle is 6.5 times the exterior angle. Let interior be I, exterior be E. So I = 6.5E. Since I + E = 180, substituting gives 6.5E + E = 7.5E = 180. E = 180 / 7.5 = 24. That seems right. Then n = 360 / E = 360 / 24 = 15.
</execution_1.1>

<execution_1.2>
So:

156 = [(n - 2)/n] * 180

Multiply both sides by n:

156n = 180(n - 2)

Expand the right side:

156n = 180n - 360

Subtract 156n from both sides:

0 = 24n - 360

Add 360 to both sides:

24n = 360

Divide both sides by 24:

n = 15
</execution_1.2>

<execution_1.3>
Since they are supplementary, that's the only way. So the relationship is correctly set up. So I think this is solid. No mistakes here.
</execution_1.3>
</Execution_1>

<Final_answer>
Therefore, the polygon has $\boxed{15}$ sides.
</Final_answer>
</think>