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Question:

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 \(360°\).)

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.