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Difficulty: MediumTriangle Properties and Angle Theorems

In △ABC\triangle ABC, the measures of the interior angles ∠A\angle A, ∠B\angle B, and ∠C\angle C are in the ratio 3:4:53:4:5, respectively. What is the measure of the largest exterior angle of △ABC\triangle ABC?

  1. A
    45∘45^\circ
  2. B
    75∘75^\circ
  3. C
    105∘105^\circ
  4. D
    108∘108^\circ
  5. 135∘135^\circAnswer

Answer

135 degrees
The interior angles of a triangle sum to 180∘180^\circ. Since the angles are in the ratio 3:4:53:4:5, they can be represented as 3x3x, 4x4x, and 5x5x. Adding these gives 12x=180∘12x = 180^\circ, which simplifies to x=15∘x = 15^\circ. The interior angles are therefore 45∘45^\circ, 60∘60^\circ, and 75∘75^\circ. Because an exterior angle is supplementary to its adjacent interior angle, the largest exterior angle is paired with the smallest interior angle: 180∘−45∘=135∘180^\circ - 45^\circ = 135^\circ. Alternatively, the exterior angle is equal to the sum of the two non-adjacent interior angles: 60∘+75∘=135∘60^\circ + 75^\circ = 135^\circ.

Step-by-Step Solution

1
Represent the interior angles algebraically using the given ratio.
Let the measures of the interior angles be 3x3x, 4x4x, and 5x5x.
The ratio of the angles is 3:4:53:4:5, so their measures must be multiples of these ratio numbers by the same factor xx.
2
Set up and solve an equation for xx using the triangle angle sum theorem.
3x+4x+5x=180∘  ⟹  12x=180∘  ⟹  x=15∘3x + 4x + 5x = 180^\circ \implies 12x = 180^\circ \implies x = 15^\circ.
The sum of the interior angles of any triangle is always 180∘180^\circ.
3
Determine the measures of the three interior angles.
The angles measure 3(15∘)=45∘3(15^\circ) = 45^\circ, 4(15∘)=60∘4(15^\circ) = 60^\circ, and 5(15∘)=75∘5(15^\circ) = 75^\circ.
Multiplying the value of xx by each term of the ratio gives the individual interior angle measures.
4
Find the largest exterior angle of the triangle.
The largest exterior angle is supplementary to the smallest interior angle: 180∘−45∘=135∘180^\circ - 45^\circ = 135^\circ.
An exterior angle and its adjacent interior angle form a linear pair and sum to 180∘180^\circ. The smallest interior angle will yield the largest exterior angle.

Key Concept

Triangle Angle Sum Theorem and Exterior Angle Relationships

Alternative Method

The exterior angle at any vertex of a triangle is equal to the sum of the measures of the two opposite interior angles. The two largest interior angles are 60∘60^\circ and 75∘75^\circ. Therefore, the largest exterior angle is the sum of these two angles: 60∘+75∘=135∘60^\circ + 75^\circ = 135^\circ.
Estimated Time:1m 0s
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Topics

Triangle Properties and Angle Theorems

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