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Difficulty: HardAngles, Parallel Lines, and Polygons

Three of the interior angles of a convex polygon are each 120∘120^\circ, while the remaining interior angles are each 160∘160^\circ. How many sides does the polygon have?

Answer: 12 sides

Answer

The polygon has 12 sides.
The sum of the interior angles of an nn-sided polygon is (n−2)×180∘(n - 2) \times 180^\circ. From the problem, the sum of the angles is 3×120∘+(n−3)×160∘=160∘n−120∘3 \times 120^\circ + (n - 3) \times 160^\circ = 160^\circ n - 120^\circ. Equating 180∘(n−2)180^\circ(n - 2) to 160∘n−120∘160^\circ n - 120^\circ gives 180∘n−360∘=160∘n−120∘180^\circ n - 360^\circ = 160^\circ n - 120^\circ, which simplifies to 20∘n=240∘20^\circ n = 240^\circ, yielding n=12n = 12.

Step-by-Step Solution

1
Write down the standard formula for the sum of interior angles of an nn-sided convex polygon.
Sum of interior angles = (n−2)×180∘(n - 2) \times 180^\circ.
Any nn-sided convex polygon can be split into (n−2)(n-2) triangles, each having an interior angle sum of 180∘180^\circ.
2
Express the total sum of interior angles using the given values.
Sum = 3(120∘)+(n−3)(160∘)=360∘+160∘n−480∘=160∘n−120∘3(120^\circ) + (n - 3)(160^\circ) = 360^\circ + 160^\circ n - 480^\circ = 160^\circ n - 120^\circ.
Three angles are 120∘120^\circ, so the remaining (n−3)(n - 3) angles must each equal 160∘160^\circ.
3
Equate the theoretical sum to the calculated sum.
180∘(n−2)=160∘n−120∘  ⟹  180∘n−360∘=160∘n−120∘180^\circ(n - 2) = 160^\circ n - 120^\circ \implies 180^\circ n - 360^\circ = 160^\circ n - 120^\circ.
Both expressions represent the total interior angle sum of the same polygon.
4
Solve the equation for the number of sides nn.
20∘n=240∘  ⟹  n=1220^\circ n = 240^\circ \implies n = 12.
Subtracting 160∘n160^\circ n from both sides and adding 360∘360^\circ yields 20∘n=240∘20^\circ n = 240^\circ.

Key Concept

Sum of Interior Angles of a Polygon
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Topics

Angles, Parallel Lines, and Polygons

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