If the measure of the interior angle of a regular polygon is 100° greater than the measure of its exterior angle then how many sides does it have?

Options

A. 10
B. 12
C. 9
D. 15

Answer

To solve the problem, we need to establish the relationship between the interior angle and the exterior angle of a regular polygon.

### Step 1: Understand the relationship between interior and exterior angles
The interior angle (I) and the exterior angle (E) of a polygon are related by the formula:
\[ I + E = 180^\circ \]

### Step 2: Set up the equation based on the problem statement
According to the problem, the interior angle is 100° greater than the exterior angle:
\[ I = E + 100^\circ \]

### Step 3: Substitute the expression for I into the first equation
Now, we can substitute the expression for I from step 2 into the equation from step 1:
\[ (E + 100^\circ) + E = 180^\circ \]

### Step 4: Simplify the equation
Combine like terms:
\[ 2E + 100^\circ = 180^\circ \]

### Step 5: Solve for E
Subtract 100° from both sides:
\[ 2E = 80^\circ \]
Now divide by 2:
\[ E = 40^\circ \]

### Step 6: Find the number of sides of the polygon
The exterior angle of a regular polygon can also be calculated using the formula:
\[ E = \frac{360^\circ}{n} \]
where n is the number of sides of the polygon. We can set this equal to the value we found for E:
\[ \frac{360^\circ}{n} = 40^\circ \]

### Step 7: Solve for n
To find n, we can cross-multiply:
\[ 360^\circ = 40^\circ \times n \]
Now divide both sides by 40°:
\[ n = \frac{360^\circ}{40^\circ} = 9 \]

### Conclusion
The polygon has **9 sides**.

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