Question

question
in \\( \triangle k l m, m \angle k = 24 ^ { \circ } \\) and \\( m \angle l = 135 ^ { \circ } \\). which statement about the sides of \\( \triangle k l m \\) must be true?
answer
\\( m k < l m < k l \\)
\\( k l < m k < l m \\)
\\( l m < m k < k l \\)
\\( m k < k l < l m \\)
\\( l m < k l < m k \\)
\\( k l < l m < m k \\)

Explanation:

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

The sum of angles in a triangle is \(180^{\circ}\). Given \(\angle K = 24^{\circ}\) and \(\angle L=135^{\circ}\), then \(\angle M=180^{\circ}-(24^{\circ} + 135^{\circ})=180^{\circ}-159^{\circ} = 21^{\circ}\).

Step2: Apply the angle - side relationship

In a triangle, the larger the angle, the longer the side opposite it. Since \(\angle L(135^{\circ})>\angle K(24^{\circ})>\angle M(21^{\circ})\), the sides opposite these angles follow the same order. The side opposite \(\angle L\) is \(MK\), the side opposite \(\angle K\) is \(LM\), and the side opposite \(\angle M\) is \(KL\). So \(MK>LM>KL\), which can be rewritten as \(KL < LM < MK\).

Answer:

\(KL < LM < MK\)