Question

The interior angle of a regular polygon exceeds the exterior angle by \(132^\circ\). Then the number of sides in the polygon is

Answer 1

Correct Answer: 1

Step 1: Understanding the Question:
The problem asks us to determine the total number of sides of a given regular polygon.
We are provided with a specific relationship between its angles.
The interior angle of this regular polygon is stated to be exactly 132$^\circ$ larger than its corresponding exterior angle.
Step 2: Key Formula or Approach:
For any convex polygon, the sum of an interior angle and its corresponding adjacent exterior angle is always exactly 180$^\circ$ because they lie on a straight line.
The sum of all exterior angles of any convex polygon is always 360$^\circ$.
For a regular polygon with $n$ sides, each exterior angle is equal to $\frac{360^\circ}{n}$.
By finding the exact value of the exterior angle, we can simply divide 360$^\circ$ by that value to find the number of sides.
Step 3: Detailed Explanation:

Let the measure of the interior angle of the regular polygon be $I$.

Let the measure of the adjacent exterior angle be $E$.

According to the fundamental property of polygons, these two angles form a linear pair.

Therefore, we can write the first equation: \[ I + E = 180^\circ \]

The problem states that the interior angle exceeds the exterior angle by exactly 132$^\circ$.

This gives us our second equation: \[ I - E = 132^\circ \]

We now have a straightforward system of two linear equations.

We can subtract the second equation from the first to eliminate $I$ and solve directly for $E$.

\[ (I + E) - (I - E) = 180^\circ - 132^\circ \]

\[ 2E = 48^\circ \]

Dividing both sides by 2 gives the value of the exterior angle.

\[ E = 24^\circ \]

Now that we know the exterior angle is exactly 24$^\circ$, we can find the number of sides, denoted as $n$.

The formula connecting the number of sides to the exterior angle of a regular polygon is $n = \frac{360^\circ}{E}$.

Substitute the value of $E$ into the formula.

\[ n = \frac{360}{24} \]

We can simplify this division by dividing both numbers by 12 first.

\[ n = \frac{30}{2} \]

\[ n = 15 \]

The regular polygon has exactly 15 sides.

Step 4: Final Answer:
The number of sides in the polygon is 15.