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Difficulty: HardPolygon Angles and Properties

An irregular convex hexagon has two interior angles measuring 90∘90^\circ and 130∘130^\circ, respectively. The remaining four interior angles have measures in the ratio 5:6:7:75:6:7:7. What is the degree measure of the largest interior angle in this hexagon?

  1. A
    100∘100^\circ
  2. B
    120∘120^\circ
  3. C
    130∘130^\circ
  4. 140∘140^\circAnswer
  5. E
    150∘150^\circ

Answer

The correct answer is 140∘140^\circ.
The sum of the interior angles of a hexagon is 720∘720^\circ. Subtracting the two given angles (90∘90^\circ and 130∘130^\circ) leaves 500∘500^\circ for the remaining four angles. Since these angles are in the ratio 5:6:7:75:6:7:7, we represent them as 5x5x, 6x6x, 7x7x, and 7x7x. Their sum is 25x=500∘25x = 500^\circ, which gives x=20∘x = 20^\circ. The largest of the remaining angles is 7x=7(20∘)=140∘7x = 7(20^\circ) = 140^\circ. Since 140∘140^\circ is greater than both 90∘90^\circ and 130∘130^\circ, it is the largest interior angle of the hexagon.

Step-by-Step Solution

1
Calculate the sum of the interior angles of a hexagon.
Using the formula for the sum of the interior angles of a polygon with nn sides, (n−2)×180∘(n - 2) \times 180^\circ, for a hexagon (n=6n = 6), the sum is (6−2)×180∘=4×180∘=720∘(6 - 2) \times 180^\circ = 4 \times 180^\circ = 720^\circ.
To establish the total sum of all interior angles of the polygon.
2
Subtract the two known angle measures from the total sum.
720∘−90∘−130∘=500∘720^\circ - 90^\circ - 130^\circ = 500^\circ.
To find the sum of the remaining four interior angles.
3
Set up an algebraic equation to find the value of one ratio unit, xx.
5x+6x+7x+7x=500∘  ⟹  25x=500∘  ⟹  x=20∘5x + 6x + 7x + 7x = 500^\circ \implies 25x = 500^\circ \implies x = 20^\circ.
To determine the constant multiplier for the ratio of the remaining angles.
4
Calculate the measures of the remaining angles and determine the largest angle.
The remaining angles are 5(20∘)=100∘5(20^\circ) = 100^\circ, 6(20∘)=120∘6(20^\circ) = 120^\circ, 7(20∘)=140∘7(20^\circ) = 140^\circ, and 7(20∘)=140∘7(20^\circ) = 140^\circ. Comparing all six angles of the hexagon (90∘,100∘,120∘,130∘,140∘,140∘90^\circ, 100^\circ, 120^\circ, 130^\circ, 140^\circ, 140^\circ), the largest angle is 140∘140^\circ.
To identify the maximum interior angle measure of the hexagon.

Key Concept

The sum of the interior angles of a convex polygon with nn sides is (n−2)×180∘(n - 2) \times 180^\circ. The individual angle measures in an irregular polygon can be determined using algebraic representations of their relationships or ratios.

Alternative Method

Once the value of the ratio unit x=20∘x = 20^\circ is determined, you can quickly find the largest candidate angle by multiplying the largest ratio component (77) by xx to get 7(20∘)=140∘7(20^\circ) = 140^\circ, and then compare it to the given angles (90∘90^\circ and 130∘130^\circ) to verify that it is indeed the largest.
Estimated Time:1m 30s
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Topics

Polygon Angles and Properties

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