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Difficulty: MediumTriangle Properties and Angle Theorems

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

  1. A
    90∘90^\circ
  2. B
    108∘108^\circ
  3. C
    126∘126^\circ
  4. 144∘144^\circAnswer
  5. E
    150∘150^\circ

Answer

144∘144^\circ
The sum of the interior angles of a triangle is 180∘180^\circ. Given the ratio 2:3:52:3:5, the angles are 2x2x, 3x3x, and 5x5x. Adding these gives 10x=180∘10x = 180^\circ, which solves to x=18∘x = 18^\circ. The smallest interior angle is 2(18∘)=36∘2(18^\circ) = 36^\circ. Since an exterior angle is supplementary to its adjacent interior angle, the largest exterior angle is the supplement of the smallest interior angle: 180∘−36∘=144∘180^\circ - 36^\circ = 144^\circ.

Step-by-Step Solution

1
Represent the three interior angles of the triangle in terms of a variable xx based on the given ratio.
The angles can be expressed as 2x2x, 3x3x, and 5x5x.
A ratio of 2:3:52:3:5 means the angles are multiples of a common factor xx.
2
Set up and solve an equation for xx using the triangle angle sum theorem.
2x+3x+5x=180∘  ⟹  10x=180∘  ⟹  x=18∘2x + 3x + 5x = 180^\circ \implies 10x = 180^\circ \implies x = 18^\circ.
The sum of the interior angles of any triangle is always 180∘180^\circ.
3
Determine the measures of all three interior angles.
The angles are 2(18∘)=36∘2(18^\circ) = 36^\circ, 3(18∘)=54∘3(18^\circ) = 54^\circ, and 5(18∘)=90∘5(18^\circ) = 90^\circ.
This identifies the individual interior angle measures to find the smallest one.
4
Calculate the measure of the largest exterior angle by finding the supplement of the smallest interior angle.
180∘−36∘=144∘180^\circ - 36^\circ = 144^\circ.
An interior angle and its adjacent exterior angle form a linear pair and sum to 180∘180^\circ. To maximize the exterior angle, we must subtract the smallest interior angle.

Key Concept

Interior and exterior angle relationships in triangles and ratio partitioning

Alternative Method

By the Exterior Angle Theorem, an exterior angle of a triangle equals the sum of the two non-adjacent interior angles. To find the largest exterior angle, sum the two largest interior angles. The two largest interior angles in ratio units are 33 parts and 55 parts, summing to 88 parts. Since the total sum of the interior angles is 1010 parts (180∘180^\circ), each part is 18∘18^\circ. Thus, the largest exterior angle is 8×18∘=144∘8 \times 18^\circ = 144^\circ.
Estimated Time:1m 0s
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Topics

Triangle Properties and Angle Theorems

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