Question

in the diagram below of triangle klm, n is the midpoint of \\( \overline{km} \\) and o is the midpoint of \\( \overline{lm} \\). if m\\( \angle mlk = 8x + 86 \\), and m\\( \angle mon = 142 - 6x \\), what is the measure of \\( \angle mlk \\)?

Explanation:

Step1: Use the mid - segment theorem

Since \(N\) is the midpoint of \(\overline{KM}\) and \(O\) is the midpoint of \(\overline{LM}\), by the mid - segment theorem, \(NO\parallel KL\). When two parallel lines are cut by a transversal (\(ML\) in this case), \(\angle MON+\angle MLK = 180^{\circ}\) (they are same - side interior angles).

Step2: Set up the equation

We know that \(m\angle MLK=8x + 86\) and \(m\angle MON=142-6x\). Substitute into the equation \((8x + 86)+(142-6x)=180\).
Simplify the left - hand side:

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Subtract \(228\) from both sides: \(2x=180 - 228=-48\).
Divide both sides by \(2\): \(x=-24\).

Step3: Find the measure of \(\angle MLK\)

Substitute \(x =-24\) into the formula for \(m\angle MLK\).
\(m\angle MLK=8x + 86\).

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This is wrong. Wait, we made a mistake. Since \(NO\parallel KL\), \(\angle MLK=\angle MON\) (corresponding angles, if we consider the correct parallel - line transversal relationship. The mid - segment \(NO\) is parallel to \(KL\) and the transversal \(ML\) gives \(\angle MLK\) and \(\angle MON\) as corresponding angles).
Set \(8x + 86=142-6x\).
Add \(6x\) to both sides: \(8x+6x + 86=142-6x+6x\), \(14x+86=142\).
Subtract \(86\) from both sides: \(14x=142 - 86 = 56\).
Divide both sides by \(14\): \(x = 4\).

Step4: Calculate \(m\angle MLK\)

Substitute \(x = 4\) into \(m\angle MLK=8x + 86\).
\(m\angle MLK=8\times4+86=32 + 86=118\).

Answer:

\(118\)