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Difficulty: MediumAngles, Parallel Lines, and Polygons

The measures of the exterior angles of an convex hexagon are given as x∘x^\circ, (x+10)∘(x + 10)^\circ, (2x−5)∘(2x - 5)^\circ, (x+25)∘(x + 25)^\circ, (2x+15)∘(2x + 15)^\circ, and (x−5)∘(x - 5)^\circ. What is the measure of the largest interior angle of the hexagon?

  1. A
    95∘95^\circ
  2. B
    85∘85^\circ
  3. 145∘145^\circAnswer
  4. D
    135∘135^\circ

Answer

145∘145^\circ
The sum of all exterior angles of a convex polygon is 360∘360^\circ. Summing the given expressions yields 8x+40=360∘8x + 40 = 360^\circ, which gives x=40∘x = 40^\circ. Evaluating each exterior angle shows that the smallest exterior angle is (40−5)∘=35∘(40 - 5)^\circ = 35^\circ. Since interior and exterior angles on a straight line are supplementary, the largest interior angle is 180∘−35∘=145∘180^\circ - 35^\circ = 145^\circ.

Step-by-Step Solution

1
Set up equation using the sum of exterior angles
x+(x+10)+(2x−5)+(x+25)+(2x+15)+(x−5)=360∘x + (x + 10) + (2x - 5) + (x + 25) + (2x + 15) + (x - 5) = 360^\circ
The sum of the exterior angles of any convex polygon is always 360∘360^\circ.
2
Simplify and solve for xx
8x+40=360  ⟹  8x=320  ⟹  x=40∘8x + 40 = 360 \implies 8x = 320 \implies x = 40^\circ
Combining like terms gives 8x+40=3608x + 40 = 360.
3
Find the smallest exterior angle
x−5=40−5=35∘x - 5 = 40 - 5 = 35^\circ
The largest interior angle corresponds to the smallest exterior angle because an interior angle and its adjacent exterior angle add up to 180∘180^\circ.
4
Calculate the largest interior angle
180∘−35∘=145∘180^\circ - 35^\circ = 145^\circ
Subtracting the smallest exterior angle from 180∘180^\circ yields the largest interior angle.

Key Concept

Exterior and Interior Angle Relationship in Polygons
Estimated Time:1m 30s
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Topics

Angles, Parallel Lines, and Polygons

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