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Difficulty: HardPolygon Angles and Properties

A convex polygon has nn sides. The sum of the measures of all but one of its interior angles is 2020∘2020^\circ. What is the measure, in degrees, of the remaining interior angle?

Answer: 140 degrees

Answer

The measure of the remaining interior angle is 140∘140^\circ.
For a convex polygon with nn sides, the sum of all interior angles is (n−2)×180∘(n-2) \times 180^\circ. If we represent the remaining interior angle as xx, then the sum of all interior angles can be written as 2020∘+x2020^\circ + x. Because the polygon is convex, the measure of the remaining interior angle must satisfy the inequality 0∘<x<180∘0^\circ < x < 180^\circ. Substituting this into the sum expression gives the inequality 2020∘<(n−2)×180∘<2200∘2020^\circ < (n-2) \times 180^\circ < 2200^\circ. Dividing by 180∘180^\circ, we find 11.22<n−2<12.2211.22 < n-2 < 12.22. Since n−2n-2 must be an integer, n−2n-2 must equal 1212, which means the total sum of the interior angles is 12×180∘=2160∘12 \times 180^\circ = 2160^\circ. The remaining angle is found by subtracting 2020∘2020^\circ from 2160∘2160^\circ, resulting in 140∘140^\circ.

Step-by-Step Solution

1
Use the polygon interior angle sum formula for an nn-sided polygon.
The sum of all interior angles is (n−2)×180∘(n-2) \times 180^\circ.
This formula connects the number of sides of a convex polygon to the total sum of its interior angles.
2
Set up an equation containing the sum of all but one angle (2020∘2020^\circ) and the remaining angle (xx).
(n−2)×180∘=2020∘+x(n-2) \times 180^\circ = 2020^\circ + x
The total sum of all interior angles is equal to the sum of the n−1n-1 known angles plus the remaining angle.
3
Apply the convexity constraint 0∘<x<180∘0^\circ < x < 180^\circ to construct an inequality for the total sum of the interior angles.
2020∘<(n−2)×180∘<2200∘2020^\circ < (n-2) \times 180^\circ < 2200^\circ
Since the remaining angle must be strictly between 0∘0^\circ and 180∘180^\circ for a convex polygon, adding 2020∘2020^\circ gives the boundaries for the total sum.
4
Divide the inequality by 180∘180^\circ to isolate the term n−2n-2.
11.22<n−2<12.2211.22 < n-2 < 12.22
This determines the numerical boundaries for the integer value of n−2n-2.
5
Find the unique integer value for n−2n-2 and calculate the exact total sum of the interior angles.
n−2=12n-2 = 12, which gives a total sum of 12×180∘=2160∘12 \times 180^\circ = 2160^\circ.
Because nn must be an integer, n−2n-2 must be an integer. The only integer in the interval (11.22,12.22)(11.22, 12.22) is 1212.
6
Subtract the sum of the other angles from the total sum of the interior angles to solve for xx.
x=2160∘−2020∘=140∘x = 2160^\circ - 2020^\circ = 140^\circ.
This yields the exact value of the remaining interior angle.

Key Concept

The sum of the interior angles of an nn-sided convex polygon is (n−2)×180∘(n-2) \times 180^\circ, where each interior angle is strictly between 0∘0^\circ and 180∘180^\circ.
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Topics

Polygon Angles and Properties

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