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Difficulty: MediumPolygon Angles and Properties

The measures of the interior angles of a convex pentagon are in the ratio 2:3:4:4:52:3:4:4:5. What is the degree measure of the smallest interior angle in this pentagon?

  1. A
    40∘40^\circ
  2. 60∘60^\circAnswer
  3. C
    80∘80^\circ
  4. D
    108∘108^\circ
  5. E
    150∘150^\circ

Answer

60∘60^\circ
The sum of the interior angles of a convex pentagon is calculated as (5−2)×180∘=540∘(5 - 2) \times 180^\circ = 540^\circ. Representing the five angle measures in the given ratio as 2x2x, 3x3x, 4x4x, 4x4x, and 5x5x, their sum is 18x=540∘18x = 540^\circ, which gives x=30∘x = 30^\circ. The smallest angle corresponds to the smallest term in the ratio, which is 2x=2×30∘=60∘2x = 2 \times 30^\circ = 60^\circ. Therefore, the correct measure is 60∘60^\circ.

Step-by-Step Solution

1
Determine the sum of the interior angles of a convex pentagon.
The sum is (5−2)×180∘=3×180∘=540∘(5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ.
The sum of the interior angles of any convex nn-gon is given by the formula (n−2)×180∘(n - 2) \times 180^\circ, and a pentagon has 55 sides.
2
Set up an algebraic equation using the given ratio of the angle measures.
Let the angle measures be 2x2x, 3x3x, 4x4x, 4x4x, and 5x5x. Their sum is 2x+3x+4x+4x+5x=18x=540∘2x + 3x + 4x + 4x + 5x = 18x = 540^\circ.
The sum of the actual angle measures must equal the total interior angle sum of the pentagon.
3
Solve for the variable xx.
x=540∘/18=30∘x = 540^\circ / 18 = 30^\circ.
Dividing the total sum by the sum of the ratio parts gives the value of a single ratio unit.
4
Calculate the measure of the smallest interior angle.
Smallest angle =2x=2×30∘=60∘= 2x = 2 \times 30^\circ = 60^\circ.
The smallest term in the ratio is 22, so multiplying this term by xx gives the smallest angle 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. For an irregular polygon with angles in a given ratio, the individual angle measures can be found by setting up a linear equation where the sum of the ratio parts multiplied by a variable equals the total sum.
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Topics

Polygon Angles and Properties

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