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Difficulty: MediumPolygon Angles and Properties

The sum of the measures of all but one of the interior angles of a convex polygon is 2010∘2010^\circ. What is the measure, in degrees, of the remaining interior angle?

Answer: 150 degrees

Answer

The measure of the remaining interior angle is 150 degrees.
The sum of the interior angles of any convex polygon with nn sides is a multiple of 180∘180^\circ given by (n−2)×180∘(n-2) \times 180^\circ. Because the polygon is convex, the measure of the remaining angle must be strictly less than 180∘180^\circ. Thus, the total sum of all interior angles must be the smallest multiple of 180∘180^\circ that is strictly greater than the given sum of 2010∘2010^\circ. Since 11×180∘=1980∘11 \times 180^\circ = 1980^\circ (which is less than 2010∘2010^\circ), the total sum must be at least 12×180∘=2160∘12 \times 180^\circ = 2160^\circ. Subtracting the given sum of the other angles from this total gives 2160∘−2010∘=150∘2160^\circ - 2010^\circ = 150^\circ. Since 150∘150^\circ is less than 180∘180^\circ, this is a mathematically valid remaining angle for a convex polygon.

Step-by-Step Solution

1
Set up the inequality for the sum of the interior angles.
The total sum S=(n−2)×180∘S = (n-2) \times 180^\circ must satisfy 2010∘<S<2010∘+180∘2010^\circ < S < 2010^\circ + 180^\circ, which simplifies to 2010∘<S<2190∘2010^\circ < S < 2190^\circ.
Since the polygon is convex, the remaining interior angle must have a measure strictly between 0∘0^\circ and 180∘180^\circ.
2
Determine the value of n−2n-2 by finding the unique integer multiple.
Dividing the inequality by 180∘180^\circ gives 11.17<n−2<12.1711.17 < n-2 < 12.17. Since nn must be an integer, n−2=12n-2 = 12, which means the polygon has n=14n = 14 sides.
The number of sides of a polygon must be a whole number, so n−2n-2 must be an integer.
3
Calculate the measure of the remaining interior angle.
x=(12×180∘)−2010∘=2160∘−2010∘=150∘x = (12 \times 180^\circ) - 2010^\circ = 2160^\circ - 2010^\circ = 150^\circ.
Subtract the sum of the other interior angles from the total sum of the interior angles of a 14-gon.

Key Concept

The sum of the interior angles of a convex polygon with nn sides is (n−2)×180∘(n-2) \times 180^\circ, and each interior angle of a convex polygon must measure strictly less than 180∘180^\circ.
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Topics

Polygon Angles and Properties

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