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Difficulty: MediumQuadrilaterals and Polygons

In a convex polygon, the sum of the measures of all interior angles except one is 1,200∘1,200^\circ. If the measure of the remaining interior angle is an integer degree, what is the number of sides of the polygon?

  1. A
    7
  2. B
    8
  3. 9Answer
  4. D
    10
  5. E
    11

Answer

The number of sides of the polygon is 9.
The interior angle sum of an nn-sided polygon is (n−2)×180∘(n - 2) \times 180^\circ. Because the polygon is convex, the missing angle xx must be between 0∘0^\circ and 180∘180^\circ. Adding xx to 1,200∘1,200^\circ gives a total sum between 1,200∘1,200^\circ and 1,380∘1,380^\circ. The only multiple of 180∘180^\circ in this range is 1,260∘1,260^\circ, which corresponds to (n−2)=7(n - 2) = 7, or n=9n = 9 sides.

Step-by-Step Solution

1
Set up the inequality for the interior angle sum of a convex polygon
The sum of the interior angles of an nn-sided convex polygon is S=(n−2)×180∘S = (n - 2) \times 180^\circ. Let xx be the measure of the remaining interior angle, where 0∘<x<180∘0^\circ < x < 180^\circ.
Every interior angle of a convex polygon must be strictly greater than 0∘0^\circ and strictly less than 180∘180^\circ.
2
Formulate the bounds for (n−2)×180∘(n - 2) \times 180^\circ
Since S=1,200∘+xS = 1,200^\circ + x, we have 1,200∘<(n−2)×180∘<1,200∘+180∘=1,380∘1,200^\circ < (n - 2) \times 180^\circ < 1,200^\circ + 180^\circ = 1,380^\circ.
Adding the bounds of xx (0∘0^\circ to 180∘180^\circ) to the given sum of 1,200∘1,200^\circ establishes the valid range for the total angle sum.
3
Solve for the integer value of nn
Dividing the inequality by 180∘180^\circ gives 6.67<n−2<7.676.67 < n - 2 < 7.67. Since n−2n - 2 must be an integer, n−2=7n - 2 = 7, which means n=9n = 9.
The only integer in the range (6.67,7.67)(6.67, 7.67) is 7, corresponding to a 9-sided polygon with a remaining angle of x=1,260∘−1,200∘=60∘x = 1,260^\circ - 1,200^\circ = 60^\circ.

Key Concept

Polygon Interior Angle Sum Theorem

Alternative Method

Divide 1,200∘1,200^\circ by 180∘180^\circ to get 6.676.67. Since (n−2)(n - 2) must be an integer and the remaining angle is positive and less than 180∘180^\circ, round 6.676.67 up to the next integer, 7. Thus, n−2=7n - 2 = 7, so n=9n = 9.
Estimated Time:1m 15s
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Topics

Quadrilaterals and Polygons

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