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Difficulty: Very hardPolygon Angles and Properties

The measures of five of the interior angles of a convex hexagon are in the ratio 3:4:5:6:73:4:5:6:7. The measure of the sixth interior angle is 30∘30^\circ less than the average measure of the other five angles. What is the degree measure of the largest interior angle of the hexagon?

  1. A
    125∘125^\circ
  2. B
    161∘161^\circ
  3. 175∘175^\circAnswer
  4. D
    202∘202^\circ
  5. E
    203∘203^\circ

Answer

The correct answer is 175^\circ because solving the equation for the sum of the hexagon's interior angles yields a multiplier of x=25x = 25, making the largest angle 7x=175∘7x = 175^\circ.
The correct answer is 175° because the sum of the interior angles of a hexagon is (6−2)×180∘=720∘(6-2) \times 180^\circ = 720^\circ. Representing the five ratio-based angles as 3x3x, 4x4x, 5x5x, 6x6x, and 7x7x gives their sum as 25x25x and their average as 5x5x. The sixth angle is therefore 5x−305x - 30. Setting up the sum of all six angles yields 25x+(5x−30)=720  ⟹  30x=750  ⟹  x=2525x + (5x - 30) = 720 \implies 30x = 750 \implies x = 25. The largest angle is 7x=7(25)=175∘7x = 7(25) = 175^\circ.

Step-by-Step Solution

1
Find the sum of the interior angles of a hexagon.
Sum = 720∘720^\circ
Using the interior angle sum formula S=(n−2)×180∘S = (n - 2) \times 180^\circ for a hexagon where n=6n = 6, we get S=(6−2)×180∘=4×180∘=720∘S = (6 - 2) \times 180^\circ = 4 \times 180^\circ = 720^\circ.
2
Express the five ratio-based angles in terms of a variable xx.
The angles are 3x3x, 4x4x, 5x5x, 6x6x, and 7x7x.
Since the measures of five angles are in the ratio 3:4:5:6:73:4:5:6:7, we can define them as multiples of a common scale factor xx.
3
Compute the average measure of these five angles in terms of xx.
Average = 5x5x
The sum of the five angles is 3x+4x+5x+6x+7x=25x3x + 4x + 5x + 6x + 7x = 25x. The average is the sum divided by the count: 25x5=5x\frac{25x}{5} = 5x.
4
Express the measure of the sixth angle in terms of xx.
Sixth angle = 5x−305x - 30
The sixth angle is described as being 30∘30^\circ less than the average of the other five angles, which we found to be 5x5x.
5
Write and solve the equation for the sum of all six interior angles.
x=25x = 25
The sum of all six angles must equal the total sum of 720∘720^\circ: 25x+(5x−30)=720  ⟹  30x−30=720  ⟹  30x=750  ⟹  x=2525x + (5x - 30) = 720 \implies 30x - 30 = 720 \implies 30x = 750 \implies x = 25.
6
Calculate the measure of the largest interior angle.
175∘175^\circ
The largest interior angle corresponds to the largest term in the ratio, which is 7x7x. Substituting x=25x = 25 yields 7×25=175∘7 \times 25 = 175^\circ.

Key Concept

Applying the interior angle sum formula for polygons combined with ratio and algebraic translation properties.

Alternative Method

Instead of using the sum of interior angles, one could use the sum of exterior angles, which is always 360∘360^\circ. The exterior angles corresponding to the five angles in ratio 3:4:5:6:73:4:5:6:7 would be 180−3x180 - 3x, 180−4x180 - 4x, 180−5x180 - 5x, 180−6x180 - 6x, and 180−7x180 - 7x. The sixth exterior angle is 180−(5x−30)=210−5x180 - (5x - 30) = 210 - 5x. Summing these six exterior angles: (180−3x)+(180−4x)+(180−5x)+(180−6x)+(180−7x)+(210−5x)=1110−30x=360(180 - 3x) + (180 - 4x) + (180 - 5x) + (180 - 6x) + (180 - 7x) + (210 - 5x) = 1110 - 30x = 360. Solving for xx yields 30x=750  ⟹  x=2530x = 750 \implies x = 25. The largest interior angle corresponds to the smallest exterior angle, which is 180−7x=180−7(25)=5∘180 - 7x = 180 - 7(25) = 5^\circ, giving the largest interior angle as 180−5=175∘180 - 5 = 175^\circ.
Estimated Time:3m 0s
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Topics

Polygon Angles and Properties

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