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Difficulty: EasyTriangle Properties and Angle Theorems

The measures of the three interior angles of a triangle are in the ratio 2:3:52:3:5. What is the measure of the largest angle in the triangle?

  1. A
    36∘36^\circ
  2. B
    54∘54^\circ
  3. 90∘90^\circAnswer
  4. D
    100∘100^\circ
  5. E
    180∘180^\circ

Answer

The correct answer is 90∘90^\circ.
Since the interior angles of a triangle sum to 180∘180^\circ and their ratio is 2:3:52:3:5, the sum of the ratio parts is 2+3+5=102 + 3 + 5 = 10. Dividing the total 180∘180^\circ by 10 parts yields 18∘18^\circ per ratio unit. The largest angle corresponds to the largest part of the ratio, which is 5. Therefore, the measure of the largest angle is 5×18∘=90∘5 \times 18^\circ = 90^\circ.

Step-by-Step Solution

1
Find the total number of parts in the ratio by adding the terms together.
The total number of parts is 2+3+5=102 + 3 + 5 = 10 parts.
This determines how many equal units the total angle measure is divided into.
2
Divide the total sum of the interior angles of a triangle by the total number of parts to find the degree measure of one part.
The sum of the interior angles of a triangle is 180∘180^\circ. Dividing by the total parts gives 180∘10=18∘\frac{180^\circ}{10} = 18^\circ per part.
The Triangle Angle Sum Theorem states that the interior angles of a triangle always sum to 180∘180^\circ.
3
Multiply the value of one part by the ratio term representing the largest angle.
The largest angle is represented by the term 5, so its measure is 5×18∘=90∘5 \times 18^\circ = 90^\circ.
This gives the measure of the largest angle in the triangle.

Key Concept

The interior angles of a triangle always sum to 180∘180^\circ, and individual angle measures can be found from a given ratio by dividing the total degrees by the sum of the ratio parts.
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Topics

Triangle Properties and Angle Theorems

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