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Difficulty: MediumPolygon Angles and Properties

An irregular convex decagon (10-sided polygon) has four interior angles that each measure 150∘150^\circ. The remaining six interior angles are congruent to each other. What is the degree measure of each of these remaining six angles?

Answer: 140 degrees

Answer

The measure of each of the remaining six interior angles is 140∘140^\circ.
The total sum of the interior angles of a 10-sided convex polygon is (10−2)×180∘=1,440∘(10-2) \times 180^\circ = 1,440^\circ. Subtracting the sum of the four angles that each measure 150∘150^\circ (4×150∘=600∘4 \times 150^\circ = 600^\circ) leaves 840∘840^\circ for the remaining six angles. Since these remaining six angles are congruent, each measures 840∘÷6=140∘840^\circ \div 6 = 140^\circ.

Step-by-Step Solution

1
Calculate the total sum of the interior angles of a convex decagon.
1,440∘1,440^\circ
The interior angle sum of a polygon with nn sides is (n−2)×180∘(n-2) \times 180^\circ, and a decagon has 10 sides.
2
Find the sum of the four angles measuring 150∘150^\circ.
600∘600^\circ
Multiply the number of angles by their given degree measure.
3
Determine the sum of the remaining six congruent angles.
840∘840^\circ
Subtract the sum of the four known angles from the total interior angle sum of the decagon.
4
Divide the remaining sum by the number of congruent angles.
140∘140^\circ
Since the remaining six angles are equal in measure, dividing their sum by 6 yields the measure of each individual angle.

Key Concept

The sum of the interior angles of a convex polygon with nn sides is (n−2)×180∘(n-2) \times 180^\circ.
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Topics

Polygon Angles and Properties

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