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Difficulty: MediumAngles, Parallel Lines, and Polygons

Three of the exterior angles of a convex polygon are each 50∘50^\circ, while the remaining exterior angles are each 35∘35^\circ. How many sides does the polygon have?

  1. A
    6
  2. B
    8
  3. 9Answer
  4. D
    10

Answer

The total number of sides of the polygon is 9.
The sum of all exterior angles of a convex polygon is 360∘360^\circ. The sum of the first three angles is 3×50∘=150∘3 \times 50^\circ = 150^\circ. Subtracting this from 360∘360^\circ leaves 210∘210^\circ for the remaining angles. Dividing 210∘210^\circ by 35∘35^\circ yields 66 remaining angles. Adding the initial 33 angles gives 3+6=93 + 6 = 9 sides in total.

Step-by-Step Solution

1
State the formula for the sum of exterior angles of a convex polygon.
The sum of exterior angles is always 360∘360^\circ.
This fundamental geometric property applies to all convex polygons regardless of the number of sides.
2
Set up an algebraic equation using the given exterior angle measures.
Let kk be the number of remaining exterior angles measuring 35∘35^\circ. Then 3(50∘)+k(35∘)=360∘3(50^\circ) + k(35^\circ) = 360^\circ.
The total sum is composed of three 50∘50^\circ angles and kk remaining 35∘35^\circ angles.
3
Solve for kk.
150∘+35∘k=360∘  ⟹  35∘k=210∘  ⟹  k=6150^\circ + 35^\circ k = 360^\circ \implies 35^\circ k = 210^\circ \implies k = 6.
Subtracting 150∘150^\circ from both sides isolates the term containing kk.
4
Calculate the total number of sides nn.
n=3+6=9n = 3 + 6 = 9.
The polygon has a total number of sides equal to the total number of exterior angles (33 initial angles +6+ 6 remaining angles).

Key Concept

The sum of the exterior angles of any convex polygon is equal to 360 degrees.
Estimated Time:1m 15s
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Topics

Angles, Parallel Lines, and Polygons

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