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Difficulty: EasyPolygon Angles and Properties

For a certain four-sided convex polygon, the ratio of its interior angle measures is 1:2:3:41:2:3:4. What is the degree measure of the smallest angle?

  1. A
    18∘18^\circ
  2. B
    54∘54^\circ
  3. 36∘36^\circAnswer
  4. D
    72∘72^\circ
  5. E
    90∘90^\circ

Answer

36∘36^\circ
The correct answer is 36∘36^\circ because the sum of the interior angles of a four-sided polygon is 360∘360^\circ. The ratio 1:2:3:41:2:3:4 means the angles can be represented as xx, 2x2x, 3x3x, and 4x4x. Their sum is 10x=360∘10x = 360^\circ, which yields x=36∘x = 36^\circ. The smallest angle is xx, which is 36∘36^\circ.

Step-by-Step Solution

1
Determine the sum of the interior angles of a four-sided convex polygon.
The sum of the interior angles of a four-sided polygon is 360∘360^\circ.
The sum of the interior angles of a polygon with nn sides is given by the formula (n−2)×180∘(n-2) \times 180^\circ. For a four-sided polygon, (4−2)×180∘=360∘(4-2) \times 180^\circ = 360^\circ.
2
Find the sum of the parts of the given ratio.
1+2+3+4=101 + 2 + 3 + 4 = 10 parts
To distribute the total angle measure proportionally, the individual parts of the ratio 1:2:3:41:2:3:4 must be summed.
3
Calculate the measure of one part of the ratio.
360∘÷10=36∘360^\circ \div 10 = 36^\circ
Dividing the total sum of the interior angles by the total number of parts determines the degree measure of a single part.
4
Find the measure of the smallest angle.
1×36∘=36∘1 \times 36^\circ = 36^\circ
The smallest angle corresponds to the smallest part of the ratio, which is 1.

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 this total can be divided proportionally using a given ratio to find individual angle measures.
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Topics

Polygon Angles and Properties

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