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Difficulty: MediumAngles, Parallel Lines, and Polygons

The sum of all the interior angles of a convex polygon, except for one, is 2190∘2190^\circ. If the measure of the remaining interior angle is strictly less than 180∘180^\circ, how many sides does the polygon have?

  1. A
    13
  2. B
    14
  3. 15Answer
  4. D
    16

Answer

The polygon has 15 sides.
The total sum of interior angles of an nn-sided polygon is (n−2)×180∘(n - 2) \times 180^\circ. Given that all angles except one sum to 2190∘2190^\circ and the missing angle is between 0∘0^\circ and 180∘180^\circ, the total sum must be the smallest multiple of 180∘180^\circ greater than 2190∘2190^\circ. The next multiple of 180∘180^\circ after 2190∘2190^\circ is 2340∘2340^\circ. Dividing 2340∘2340^\circ by 180∘180^\circ gives 1313, so n−2=13n - 2 = 13, which yields n=15n = 15. The missing angle is 2340∘−2190∘=150∘2340^\circ - 2190^\circ = 150^\circ, which is valid.

Step-by-Step Solution

1
State the formula for the sum of interior angles of an nn-sided convex polygon.
S=(n−2)×180∘S = (n - 2) \times 180^\circ, where nn is an integer representing the number of sides (n≥3n \ge 3).
The sum of interior angles of any convex polygon is determined by dividing it into (n−2)(n - 2) triangles.
2
Set up an inequality using the given partial sum and the constraint on the remaining interior angle.
Let the missing interior angle be xx. Then (n−2)×180∘=2190∘+x(n - 2) \times 180^\circ = 2190^\circ + x, with 0∘<x<180∘0^\circ < x < 180^\circ.
Since the polygon is convex, every interior angle is non-reflex and greater than 0∘0^\circ.
3
Solve for the bounds of (n−2)(n - 2).
Dividing 2190∘<(n−2)×180∘<2190∘+180∘2190^\circ < (n - 2) \times 180^\circ < 2190^\circ + 180^\circ by 180∘180^\circ gives 12.166...<n−2<13.166...12.166... < n - 2 < 13.166.... Since n−2n - 2 must be an integer, n−2=13n - 2 = 13.
The total interior angle sum must be an exact integer multiple of 180∘180^\circ.
4
Calculate nn and verify the value of xx.
n=13+2=15n = 13 + 2 = 15. The total sum is 13×180∘=2340∘13 \times 180^\circ = 2340^\circ, making the remaining angle x=2340∘−2190∘=150∘x = 2340^\circ - 2190^\circ = 150^\circ.
The remaining angle 150∘150^\circ satisfies 0∘<150∘<180∘0^\circ < 150^\circ < 180^\circ, confirming n=15n = 15 is correct.

Key Concept

Sum of Interior Angles of Convex Polygons
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Topics

Angles, Parallel Lines, and Polygons

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