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Difficulty: HardPolygon Angles and Properties

An irregular convex hexagon has three interior angles that are congruent to each other, and the remaining three interior angles have measures in the ratio 3:4:53:4:5. If the sum of the measures of the three congruent angles is 360∘360^\circ, what is the degree measure of the largest interior angle of the hexagon?

  1. A
    90∘90^\circ
  2. B
    120∘120^\circ
  3. 150∘150^\circAnswer
  4. D
    250∘250^\circ
  5. E
    300∘300^\circ

Answer

The degree measure of the largest interior angle of the hexagon is 150∘150^\circ.
The correct answer is 150∘150^\circ. The total sum of the interior angles of a hexagon is (6−2)×180∘=720∘(6-2) \times 180^\circ = 720^\circ. Subtracting the sum of the three congruent angles (360∘360^\circ) leaves 360∘360^\circ for the remaining three angles. With their measures in the ratio 3:4:53:4:5, we can write the equation 3x+4x+5x=360∘3x + 4x + 5x = 360^\circ, which simplifies to 12x=360∘12x = 360^\circ, giving x=30∘x = 30^\circ. The measures of these three angles are 90∘90^\circ, 120∘120^\circ, and 150∘150^\circ. The three congruent angles each measure 360∘/3=120∘360^\circ / 3 = 120^\circ. Comparing all six angles (120∘,120∘,120∘,90∘,120∘,150∘120^\circ, 120^\circ, 120^\circ, 90^\circ, 120^\circ, 150^\circ), the largest measure is 150∘150^\circ.

Step-by-Step Solution

1
Calculate the sum of the interior angles of a convex hexagon.
720∘720^\circ
Using the formula (n−2)×180∘(n - 2) \times 180^\circ for a polygon with n=6n = 6 sides, the sum is (6−2)×180∘=4×180∘=720∘(6 - 2) \times 180^\circ = 4 \times 180^\circ = 720^\circ.
2
Determine the sum of the remaining three interior angles.
360∘360^\circ
We subtract the sum of the three congruent angles (360∘360^\circ) from the total sum of the hexagon's interior angles (720∘720^\circ): 720∘−360∘=360∘720^\circ - 360^\circ = 360^\circ.
3
Set up and solve an equation using the ratio of the remaining three angles.
x=30∘x = 30^\circ
Let the measures of the remaining three angles be 3x3x, 4x4x, and 5x5x. Their sum is 360∘360^\circ, so 3x+4x+5x=360∘  ⟹  12x=360∘  ⟹  x=30∘3x + 4x + 5x = 360^\circ \implies 12x = 360^\circ \implies x = 30^\circ.
4
Calculate the measures of all six interior angles to identify the largest one.
The angles are 120∘120^\circ, 120∘120^\circ, 120∘120^\circ, 90∘90^\circ, 120∘120^\circ, and 150∘150^\circ. The largest is 150∘150^\circ.
Each of the three congruent angles measures 360∘/3=120∘360^\circ / 3 = 120^\circ. The other three angles measure 3(30∘)=90∘3(30^\circ) = 90^\circ, 4(30∘)=120∘4(30^\circ) = 120^\circ, and 5(30∘)=150∘5(30^\circ) = 150^\circ. Comparing all these values, the maximum is 150∘150^\circ.

Key Concept

The sum of the interior angles of a convex nn-sided polygon is given by (n−2)×180∘(n-2) \times 180^\circ. Irregular polygons share this total sum, and ratio relationships can be solved by introducing a variable multiplier.
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Topics

Polygon Angles and Properties

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