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What is the number of sides of a regular polygon whose interior angles are \(156°\) each? (Remember, the sum of exterior angles of any polygon is \(360°\).)

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Question:

A
15

explanation

The sum of exterior angles of any polygon is always equal to 360 degrees, hence:

Exterior angle measurement in degrees x n = 360

where n is the number of angles (which is also the number of sides).

Also, for a given interior angle, the exterior angle = 180 – interior angle

In the problem: the exterior angle = 180 – 156 = 24 degrees

From the formula given above and measurement of the exterior angle computed: 24n = 360

\(n = 360 \div 24 = 15\)

The polygon has 15 angles and 15 sides.

An alternative approach to this problem is to derive or remember that the measurement of each angle of a regular polygon can be determined using the formula: \(\frac{\mathrm{180(n-2)} }{\mathrm{n}} = x\), where n is the number of sides and x is the measurement of the interior angle.

Substituting the given information, the equation becomes: \(\frac{\mathrm{180(n-2)} }{\mathrm{n}} = 156\), and solving for n gives 15.

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