Question

question
in $\triangle klm$, $m\angle k = 102^\circ$ and $m\angle l = 47^\circ$. which statement about the sides of $\triangle klm$ must be true?
answer
\\( \circ \\) kl > mk > lm
\\( \circ \\) mk > kl > lm
\\( \circ \\) kl > lm > mk
\\( \circ \\) mk > lm > kl
\\( \circ \\) lm > mk > kl
\\( \circ \\) lm > kl > mk

Explanation:

Step1: Find the third angle

In a triangle, the sum of interior angles is \(180^\circ\). Given \(m\angle K = 102^\circ\) and \(m\angle L=47^\circ\), let \(m\angle M=x\). Then \(102 + 47+x=180\), so \(x = 180-(102 + 47)=31^\circ\). So \(m\angle K = 102^\circ\), \(m\angle L = 47^\circ\), \(m\angle M=31^\circ\).

Step2: Relate angles to sides

In a triangle, the larger angle is opposite the longer side.

• Opposite \(\angle K\) (102°) is side \(LM\).
• Opposite \(\angle L\) (47°) is side \(MK\).
• Opposite \(\angle M\) (31°) is side \(KL\).

Since \(102^\circ>47^\circ>31^\circ\) (i.e., \(\angle K>\angle L>\angle M\)), their opposite sides follow \(LM>MK>KL\).

Answer:

\(LM > MK > KL\) (the option with this inequality)