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Question

Difficulty: MediumAngles, Parallel Lines, and Polygons

In a regular polygon, the measure of each interior angle is 132∘132^\circ greater than the measure of each exterior angle. How many sides does this polygon have?

Answer: 15 sides

Answer

The polygon has 15 sides.
Since the interior angle II and exterior angle EE of a regular polygon sum to 180∘180^\circ (I+E=180∘I + E = 180^\circ) and their given difference is I−E=132∘I - E = 132^\circ, subtracting the difference equation from the sum equation gives 2E=48∘2E = 48^\circ, which simplifies to E=24∘E = 24^\circ. The number of sides is n=360∘E=360∘24∘=15n = \frac{360^\circ}{E} = \frac{360^\circ}{24^\circ} = 15.

Step-by-Step Solution

1
Set up the linear pair equation for interior and exterior angles
I+E=180∘I + E = 180^\circ
An interior angle and its adjacent exterior angle at any vertex of a polygon lie on a straight line and sum to 180∘180^\circ.
2
Set up the given condition equation
I−E=132∘I - E = 132^\circ
The question states that each interior angle is 132∘132^\circ greater than each exterior angle.
3
Solve for the exterior angle EE
E=24∘E = 24^\circ
Subtracting I−E=132∘I - E = 132^\circ from I+E=180∘I + E = 180^\circ yields 2E=48∘2E = 48^\circ, giving E=24∘E = 24^\circ.
4
Calculate the number of sides nn
n=15n = 15
The sum of all exterior angles of any convex polygon is 360∘360^\circ, so n=360∘E=360∘24∘=15n = \frac{360^\circ}{E} = \frac{360^\circ}{24^\circ} = 15.

Key Concept

Interior and Exterior Angle Properties of Regular Polygons
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Topics

Angles, Parallel Lines, and Polygons

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