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Difficulty: MediumQuadrilaterals and Polygons

In convex pentagon ABCDEABCDE, the measure of interior angle ∠A\angle A is 100∘100^\circ. The measures of the remaining four interior angles, ∠B\angle B, ∠C\angle C, ∠D\angle D, and ∠E\angle E, are in the ratio 2:3:3:32 : 3 : 3 : 3. What is the measure, in degrees, of ∠B\angle B?

  1. A
    40∘40^\circ
  2. 80∘80^\circAnswer
  3. C
    108∘108^\circ
  4. D
    120∘120^\circ
  5. E
    160∘160^\circ

Answer

The correct answer is 80∘80^\circ, which corresponds to the measure of ∠B\angle B.
The sum of interior angles of a 5-sided polygon (pentagon) is (5−2)×180∘=540∘(5-2) \times 180^\circ = 540^\circ. Subtracting ∠A=100∘\angle A = 100^\circ leaves 440∘440^\circ for the remaining four angles. The total ratio parts for these four angles is 2+3+3+3=112 + 3 + 3 + 3 = 11. Dividing 440∘440^\circ by 11 gives 40∘40^\circ per ratio unit. Since ∠B\angle B corresponds to 2 ratio parts, its measure is 2×40∘=80∘2 \times 40^\circ = 80^\circ.

Step-by-Step Solution

1
Calculate the sum of all interior angles of the pentagon.
The sum of interior angles for an nn-sided polygon is given by (n−2)×180∘(n - 2) \times 180^\circ. For a pentagon (n=5n = 5), the sum is (5−2)×180∘=3×180∘=540∘(5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ.
Determining the total interior angle sum is necessary to find the sum of the unknown angles.
2
Subtract the known angle measure ∠A\angle A from the total interior angle sum.
The combined sum of angles ∠B+∠C+∠D+∠E=540∘−100∘=440∘\angle B + \angle C + \angle D + \angle E = 540^\circ - 100^\circ = 440^\circ.
Isolating the sum of the remaining four angles allows distribution according to the given ratio.
3
Determine the value of one ratio unit.
The sum of ratio parts is 2+3+3+3=112 + 3 + 3 + 3 = 11 parts. One part is equal to 440∘/11=40∘440^\circ / 11 = 40^\circ.
Finding the magnitude of a single ratio unit enables calculation of any individual angle.
4
Multiply the single unit value by the ratio coefficient for ∠B\angle B.
Since ∠B\angle B corresponds to 2 parts, ∠B=2×40∘=80∘\angle B = 2 \times 40^\circ = 80^\circ.
This yields the requested measure of angle ∠B\angle B.

Key Concept

Sum of interior angles of an n-sided polygon: (n−2)×180∘(n - 2) \times 180^\circ, combined with proportional partitioning of angle sums.

Alternative Method

Express the angles in terms of a variable xx. Let the remaining angles be 2x,3x,3x,3x2x, 3x, 3x, 3x. Set up the equation 100∘+2x+3x+3x+3x=540∘100^\circ + 2x + 3x + 3x + 3x = 540^\circ, simplify to 100+11x=540100 + 11x = 540, solve for x=40∘x = 40^\circ, and then evaluate ∠B=2x=80∘\angle B = 2x = 80^\circ.
Estimated Time:1m 15s
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Topics

Quadrilaterals and Polygons

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