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Difficulty: MediumAngles, Parallel Lines, and Polygons

Five of the interior angles of a convex polygon are each equal to 140∘140^\circ, while the remaining interior angles are each equal to 160∘160^\circ. Calculate the number of sides of the polygon.

Answer: 13

Answer

The number of sides of the polygon is 13.
Each 140∘140^\circ interior angle has an exterior angle of 40∘40^\circ, contributing 5×40∘=200∘5 \times 40^\circ = 200^\circ to the exterior angle sum. Each 160∘160^\circ interior angle has an exterior angle of 20∘20^\circ, contributing (n−5)×20∘(n - 5) \times 20^\circ. Since the sum of exterior angles of any convex polygon is 360∘360^\circ, setting 200+20(n−5)=360200 + 20(n - 5) = 360 yields 20n=26020n = 260, giving n=13n = 13.

Step-by-Step Solution

1
Calculate the exterior angle measures
The exterior angles are 180∘−140∘=40∘180^\circ - 140^\circ = 40^\circ (for 5 vertices) and 180∘−160∘=20∘180^\circ - 160^\circ = 20^\circ (for the remaining n−5n - 5 vertices).
Interior and exterior angles at each vertex of a polygon form a linear pair and sum to 180∘180^\circ.
2
Apply the sum of exterior angles property
5(40∘)+(n−5)(20∘)=360∘5(40^\circ) + (n - 5)(20^\circ) = 360^\circ.
The sum of exterior angles of any convex polygon is always constant and equal to 360∘360^\circ.
3
Solve the linear equation for nn
200+20n−100=360  ⟹  20n=260  ⟹  n=13200 + 20n - 100 = 360 \implies 20n = 260 \implies n = 13.
Expanding terms and isolating nn gives the exact number of sides.

Key Concept

Exterior angle sum property of convex polygons
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Topics

Angles, Parallel Lines, and Polygons

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