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Difficulty: EasyTriangle Properties and Angle Theorems

The measures of the three interior angles of a triangle are in the ratio 2:3:42:3:4. What is the measure, in degrees, of the smallest angle of the triangle?

  1. A
    20
  2. 40Answer
  3. C
    60
  4. D
    72
  5. E
    80

Answer

40
The interior angles of any triangle sum to 180∘180^\circ. Given the ratio of the angles is 2:3:42:3:4, the total number of ratio parts is 2+3+4=92 + 3 + 4 = 9. Dividing the total degree sum by the number of parts gives 180∘9=20∘\frac{180^\circ}{9} = 20^\circ per part. Since the smallest angle corresponds to the smallest part of the ratio, we multiply 22 by 20∘20^\circ to get 40∘40^\circ.

Step-by-Step Solution

1
Find the total number of parts in the ratio.
The total number of parts is 2+3+4=92 + 3 + 4 = 9.
This determines how the 180∘180^\circ total sum of a triangle's interior angles is partitioned.
2
Calculate the value of a single part of the ratio.
One part is equal to 180∘9=20∘\frac{180^\circ}{9} = 20^\circ.
The sum of the interior angles of any triangle is always 180∘180^\circ.
3
Find the measure of the smallest angle by multiplying by the smallest part of the ratio.
The smallest angle measures 2×20∘=40∘2 \times 20^\circ = 40^\circ.
The smallest angle corresponds to the smallest number in the ratio, which is 2.

Key Concept

Ratio-based angle partitioning in triangles
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Topics

Triangle Properties and Angle Theorems

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