Question

question
in △abc, m∠a = 81° and m∠b = 11°. which statement about the sides of △abc must be true?
answer
ab < bc < ca
bc < ab < ca
ca < bc < ab
bc < ca < ab
ca < ab < bc
ab < ca < bc
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Explanation:

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

In a triangle, the sum of angles is \( 180^{\circ} \). Given \( m\angle A = 81^{\circ} \) and \( m\angle B=11^{\circ} \), then \( m\angle C=180^{\circ}-(m\angle A + m\angle B)=180^{\circ}-(81^{\circ}+11^{\circ}) = 88^{\circ} \).

Step2: Use the angle - side relationship

In a triangle, the larger the angle, the longer the side opposite it. Since \( m\angle B(11^{\circ})<m\angle A(81^{\circ})<m\angle C(88^{\circ}) \), the sides opposite these angles follow the same order. The side opposite \( \angle A \) is \( BC \), the side opposite \( \angle B \) is \( AC \), and the side opposite \( \angle C \) is \( AB \). So \( AC < BC < AB \).

Answer:

\( CA < BC < AB \)