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Difficulty: MediumPolygon Angles and Properties

A convex octagon has interior angles whose measures, in degrees, are eight consecutive even integers. What is the measure, in degrees, of the largest interior angle of this octagon?

Answer: 142 degrees

Answer

The correct answer is 142.
The sum of the interior angles of a convex octagon is (8−2)×180∘=1080∘(8-2) \times 180^\circ = 1080^\circ. If we represent the eight consecutive even integer angle measures as x,x+2,x+4,x+6,x+8,x+10,x+12,x, x+2, x+4, x+6, x+8, x+10, x+12, and x+14x+14, their sum is 8x+568x + 56. Setting this equal to 1080∘1080^\circ and solving for xx yields x=128x = 128. The largest angle is x+14x + 14, which equals 128+14=142∘128 + 14 = 142^\circ.

Step-by-Step Solution

1
Calculate the sum of the interior angles of a convex octagon.
The sum of the interior angles is 1080∘1080^\circ.
The formula for the sum of the interior angles of an nn-sided polygon is (n−2)×180∘(n-2) \times 180^\circ. For an octagon (n=8n=8), the sum is (8−2)×180∘=6×180∘=1080∘(8-2) \times 180^\circ = 6 \times 180^\circ = 1080^\circ.
2
Set up an equation representing the sum of the eight consecutive even integer angle measures.
The equation is 8x+56=10808x + 56 = 1080.
Letting the smallest angle measure be xx, the eight consecutive even integer angle measures are x,x+2,x+4,x+6,x+8,x+10,x+12,x, x+2, x+4, x+6, x+8, x+10, x+12, and x+14x+14. Their sum is 8x+568x + 56, which must equal the total sum of the interior angles (1080∘1080^\circ).
3
Solve the equation for the smallest angle measure, xx.
x=128x = 128
Subtracting 56 from both sides of the equation yields 8x=10248x = 1024. Dividing both sides by 8 gives x=128x = 128.
4
Calculate the measure of the largest interior angle.
The measure of the largest angle is 142∘142^\circ.
The largest angle is represented by the expression x+14x + 14. Substituting 128128 for xx gives 128+14=142128 + 14 = 142.

Key Concept

Calculating the sum of the interior angles of a convex polygon and using algebraic methods to find unknown angle measures.
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Topics

Polygon Angles and Properties

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