## Example Questions

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### Example Question #382 : Plane Geometry

Which of the following cannot be the measure of an exterior angle of a regular polygon?      Explanation:

If one exterior angle is taken at each vertex of any convex polygon, the sum of their measures is . In a regular polygon - one with congruent sides and congruent interior angles, each exterior angle is congruent to one another. If the polygon has sides, each exterior angle has measure .

Given the common measure  Multiplying both sides by : and Since is equal to a number of sides, it is a whole number. Thus, we are looking for a value of which, when we divide 360 by it, yields a non-whole result. We see that is the correct choice, since' A quick check confirms that 360 divided by 8, 10, 12, or 15 yields a whole result.

### Example Question #11 : Other Polygons

To the nearest whole degree, give the measure of each interior angle of a regular polygon with 17 sides.      Explanation:

The measure of each interior angle of an -sided polygon can be calculated using the formula Setting : The correct choice is therefore .

### Example Question #384 : Plane Geometry

Each interior angle of a regular polygon has measure . How many sides does the polygon have?      Explanation:

The easiest way to work this is arguably to examine the exterior angles, each of which forms a linear pair with an interior angle. If an interior angle measures , then each exterior angle, which is supplementary to an interior angle, measures The measures of the exterior angles of a polygon, one per vertex, total ; in a regular polygon, they are congruent, so if there are such angles, each measures . Since the number of vertices is equal to the number of sides, if we set this equal to and solve for , we will find the number of sides. Multiply both sides by :   The polygon has 72 vertices and, thus, 72 sides.

### Example Question #11 : How To Find An Angle In A Polygon

A regular polygon has a measure of for each of its internal angles.  How many sides does it have?     Explanation:

To determine the measure of the angles of a regular polygon use:

Angle = (n – 2) x 180° / n

Thus, (n – 2) x 180° / n = 140°

180° n - 360° = 140° n

40° n = 360°

n = 360° / 40° = 9

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