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Difficulty: HardPolygon Angles and Properties

An irregular convex octagon has five interior angles that each measure 144∘144^\circ. The remaining three interior angles have measures in the ratio 3:4:53:4:5. What is the measure, in degrees, of the largest interior angle of this octagon?

Answer: 150 degrees

Answer

The measure of the largest interior angle of the octagon is 150∘150^\circ.
The total sum of the interior angles of an octagon is (8−2)×180∘=1080∘(8-2) \times 180^\circ = 1080^\circ. The sum of the five given angles is 5×144∘=720∘5 \times 144^\circ = 720^\circ, leaving a sum of 1080∘−720∘=360∘1080^\circ - 720^\circ = 360^\circ for the remaining three angles. Since these three angles are in the ratio 3:4:53:4:5, we set 3x+4x+5x=360∘3x + 4x + 5x = 360^\circ, yielding 12x=360∘12x = 360^\circ and x=30∘x = 30^\circ. The largest of these three angles is 5×30∘=150∘5 \times 30^\circ = 150^\circ. Comparing 150∘150^\circ to the other angles of the octagon (which are 144∘144^\circ, 90∘90^\circ, and 120∘120^\circ), the largest interior angle is 150∘150^\circ.

Step-by-Step Solution

1
Calculate the total sum of the interior angles of the octagon.
The total sum is 1080∘1080^\circ.
The sum of the interior angles of any convex polygon with nn sides is given by (n−2)×180∘(n - 2) \times 180^\circ. For an octagon, n=8n = 8, so the sum is (8−2)×180∘=6×180∘=1080∘(8 - 2) \times 180^\circ = 6 \times 180^\circ = 1080^\circ.
2
Calculate the sum of the five given congruent angles.
The sum of these five angles is 720∘720^\circ.
Since five angles each measure 144∘144^\circ, their combined sum is 5×144∘=720∘5 \times 144^\circ = 720^\circ.
3
Determine the sum of the remaining three interior angles.
The sum of the remaining angles is 360∘360^\circ.
Subtracting the sum of the five congruent angles from the total sum of the octagon's interior angles yields the sum of the remaining three angles: 1080∘−720∘=360∘1080^\circ - 720^\circ = 360^\circ.
4
Use the ratio 3:4:53:4:5 to find the measures of the remaining three angles.
The measures of the three angles are 90∘90^\circ, 120∘120^\circ, and 150∘150^\circ.
Let the measures of the remaining three angles be 3x3x, 4x4x, and 5x5x. Their sum is 3x+4x+5x=12x=360∘3x + 4x + 5x = 12x = 360^\circ, which gives x=30∘x = 30^\circ. The largest of these three angles is 5×30∘=150∘5 \times 30^\circ = 150^\circ.
5
Compare the measures of all interior angles of the octagon to find the largest one.
The largest angle is 150∘150^\circ.
The octagon's interior angles consist of five angles of 144∘144^\circ, and three angles of 90∘90^\circ, 120∘120^\circ, and 150∘150^\circ. Comparing these values, 150∘150^\circ is the largest measure.

Key Concept

The sum of the interior angles of a convex polygon with nn sides is (n−2)×180∘(n-2) \times 180^\circ, and ratios can be used to partition a total sum into specific parts.
Estimated Time:2m 0s
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Topics

Polygon Angles and Properties

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