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Difficulty: MediumTriangle Properties and Angle Theorems

In △ABC\triangle ABC, the lengths of the sides are AB=8AB = 8, BC=11BC = 11, and AC=14AC = 14. Which of the following inequalities correctly compares the measures of the interior angles of △ABC\triangle ABC?

  1. A
    m∠A<m∠B<m∠Cm\angle A < m\angle B < m\angle C
  2. B
    m∠B<m∠A<m∠Cm\angle B < m\angle A < m\angle C
  3. C
    m∠A<m∠C<m∠Bm\angle A < m\angle C < m\angle B
  4. m∠C<m∠A<m∠Bm\angle C < m\angle A < m\angle BAnswer
  5. E
    m∠B<m∠C<m∠Am\angle B < m\angle C < m\angle A

Answer

The correct inequality is m∠C<m∠A<m∠Bm\angle C < m\angle A < m\angle B.
In any triangle, the order of the measures of the interior angles matches the order of the lengths of their opposite sides. Since the side lengths are ordered AB<BC<ACAB < BC < AC (8<11<148 < 11 < 14), their opposite angles must be ordered in the same way. The angle opposite side ABAB is ∠C\angle C, the angle opposite side BCBC is ∠A\angle A, and the angle opposite side ACAC is ∠B\angle B. Therefore, the correct relationship is m∠C<m∠A<m∠Bm\angle C < m\angle A < m\angle B.

Step-by-Step Solution

1
Identify the theorem relating triangle side lengths to their opposite angles.
The Side-Angle Relationship Theorem states that in any triangle, the order of the measures of the angles is the same as the order of the lengths of the sides opposite to those angles.
This establishes the rule needed to compare the angle measures based on the given side lengths.
2
Determine the opposite angle for each side length in △ABC\triangle ABC.
The angle opposite side ABAB is ∠C\angle C. The angle opposite side BCBC is ∠A\angle A. The angle opposite side ACAC is ∠B\angle B.
This maps each side length to the correct angle it controls.
3
Order the side lengths and apply the mapping to order the angle measures.
Since 8<11<148 < 11 < 14, we write the side inequality as AB<BC<ACAB < BC < AC. Substituting the corresponding opposite angles gives m∠C<m∠A<m∠Bm\angle C < m\angle A < m\angle B.
This yields the final correct inequality comparing the angle measures.

Key Concept

Triangle Side-Angle Relationship Theorem

Alternative Method

Another way to solve this is to sketch the triangle to scale. By drawing the longest side AC=14AC = 14 as the horizontal base, the shortest side AB=8AB = 8 on the left, and the side BC=11BC = 11 on the right, it becomes visually apparent that the angle opposite the longest side (∠B\angle B at the top vertex) is the largest angle, and the angle opposite the shortest side (∠C\angle C at the bottom-right vertex) is the smallest angle.
Estimated Time:1m 0s
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Topics

Triangle Properties and Angle Theorems

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