Question

question in \\( \triangle a b c, m \angle a = 112 ^ { \circ } \\) and \\( m \angle b = 14 ^ { \circ } \\). which statement about the sides of \\( \triangle a b c \\) must be true? answer \\( b c < a b < c a \\) \\( a b < c a < b c \\) \\( c a < b c < a b \\) \\( b c < c a < a b \\) \\( a b < b c < c a \\) \\( c a < a b < b c \\)

Explanation:

Step1: Find the measure of angle \( C \)

The sum of angles in a triangle is \( 180^{\circ} \).
So, \( m\angle C=180^{\circ}-m\angle A - m\angle B \)
\( m\angle C = 180^{\circ}-112^{\circ}-14^{\circ}=54^{\circ} \)

Step2: Apply the angle - side relationship theorem

In a triangle, the larger the angle, the longer the side opposite it.
Since \( m\angle A(112^{\circ})>m\angle C(54^{\circ})>m\angle B(14^{\circ}) \)
The side opposite \( \angle A \) is \( BC \), the side opposite \( \angle C \) is \( AB \), and the side opposite \( \angle B \) is \( CA \)
So, \( BC > AB>CA \) (or \( CA < AB < BC \))

Answer:

CA \( < \) AB \( < \) BC