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Difficulty: HardTriangle Properties and Angle Theorems

In △ABC\triangle ABC, the measures of the three exterior angles (one at each vertex) are in the ratio 4:5:64:5:6. What is the measure of the smallest interior angle of △ABC\triangle ABC?

  1. 36∘36^\circAnswer
  2. B
    48∘48^\circ
  3. C
    84∘84^\circ
  4. D
    96∘96^\circ
  5. E
    176∘176^\circ

Answer

The correct answer is 36∘36^\circ.
The correct answer is 36∘36^\circ. The sum of the three exterior angles of a triangle is 360∘360^\circ. Given the ratio 4:5:64:5:6, we set up the equation 4k+5k+6k=360∘4k + 5k + 6k = 360^\circ, which simplifies to 15k=360∘15k = 360^\circ, yielding k=24∘k = 24^\circ. The largest exterior angle is 6(24∘)=144∘6(24^\circ) = 144^\circ. Since the interior and exterior angles at a vertex are supplementary, the smallest interior angle corresponds to the largest exterior angle, which is 180∘−144∘=36∘180^\circ - 144^\circ = 36^\circ.

Step-by-Step Solution

1
Recall the sum of the exterior angles of a triangle.
The sum of the exterior angles of any convex polygon (including a triangle) is 360∘360^\circ.
This is a fundamental geometric property of exterior angles.
2
Set up an algebraic equation to find the value of each part in the ratio.
Let the measures of the exterior angles be 4k4k, 5k5k, and 6k6k. Then: 4k+5k+6k=360∘  ⟹  15k=360∘  ⟹  k=24∘4k + 5k + 6k = 360^\circ \implies 15k = 360^\circ \implies k = 24^\circ.
The sum of the ratio parts must equal the total sum of the exterior angles.
3
Identify which exterior angle corresponds to the smallest interior angle.
Since an interior angle and its adjacent exterior angle are supplementary (180∘180^\circ sum), the smallest interior angle must correspond to the largest exterior angle, which is 6k=6(24∘)=144∘6k = 6(24^\circ) = 144^\circ.
As the exterior angle increases, the supplementary interior angle decreases.
4
Calculate the smallest interior angle.
180∘−144∘=36∘180^\circ - 144^\circ = 36^\circ.
Subtracting the largest exterior angle from 180∘180^\circ yields the smallest interior angle.

Key Concept

The sum of the exterior angles of a triangle is 360∘360^\circ, and the interior and exterior angles at each vertex are supplementary.
Estimated Time:1m 30s
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Topics

Triangle Properties and Angle Theorems

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