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Difficulty: MediumPolygon Angles and Properties

The measures of the five exterior angles of a convex pentagon are in the ratio 2:3:4:4:52:3:4:4:5. What is the measure of the largest interior angle of this pentagon?

  1. A
    80∘80^\circ
  2. B
    100∘100^\circ
  3. C
    120∘120^\circ
  4. 140∘140^\circAnswer
  5. E
    150∘150^\circ

Answer

The measure of the largest interior angle of the pentagon is 140∘140^\circ.
The correct answer is 140∘140^\circ. The sum of the exterior angles of any convex polygon is 360∘360^\circ. For a pentagon with exterior angles in the ratio 2:3:4:4:52:3:4:4:5, the sum of the ratio parts is 2+3+4+4+5=182+3+4+4+5=18 parts. Each part is equal to 360∘/18=20∘360^\circ / 18 = 20^\circ. The smallest exterior angle has 22 parts, which measures 2×20∘=40∘2 \times 20^\circ = 40^\circ. Since each interior angle is supplementary to its corresponding exterior angle, the largest interior angle corresponds to the smallest exterior angle. Thus, the largest interior angle measures 180∘−40∘=140∘180^\circ - 40^\circ = 140^\circ.

Step-by-Step Solution

1
Determine the sum of the exterior angles of a convex pentagon.
The sum of the exterior angles of any convex polygon is 360∘360^\circ.
This is a fundamental property of convex polygons and provides the total value to distribute among the ratio parts.
2
Calculate the measure of the smallest exterior angle using the given ratio of 2:3:4:4:52:3:4:4:5.
The sum of the ratio parts is 2+3+4+4+5=182+3+4+4+5=18. The smallest exterior angle has 22 parts, so its measure is 218×360∘=40∘\frac{2}{18} \times 360^\circ = 40^\circ.
The largest interior angle will be adjacent (and supplementary) to the smallest exterior angle.
3
Calculate the measure of the largest interior angle by subtracting the smallest exterior angle from 180∘180^\circ.
The largest interior angle is 180∘−40∘=140∘180^\circ - 40^\circ = 140^\circ.
An interior angle and its adjacent exterior angle are supplementary and add up to 180∘180^\circ.

Key Concept

The sum of the exterior angles of any convex polygon is 360∘360^\circ. An interior angle and its adjacent exterior angle are supplementary, meaning they add up to 180∘180^\circ. Consequently, the largest interior angle corresponds to the smallest exterior angle.
Estimated Time:1m 30s
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Topics

Polygon Angles and Properties

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