Question

find the measure of \\( \angle m p q \\).
\\( m \angle m p q= \\) \\( \circ \\)

Explanation:

Step1: Use the property of parallel lines and transversal

Since \(LM\parallel NO\), \(\angle MPQ\) and \(\angle NQP\) are same - side interior angles. But also, we can use the property that \(\angle MPQ\) and \(\angle KQO\) are related as \(\angle MPQ\) and \(\angle PQN\) are supplementary (because \(LM\parallel NO\) and \(PQ\) is a transversal, same - side interior angles). And \(\angle PQN=\angle KQO=(52 - 2x)^{\circ}\) (vertical angles). So \((10x + 28)+(52 - 2x)=180\) (supplementary angles).

Step2: Solve the equation for \(x\)

Simplify the left - hand side of the equation \(10x+28 + 52-2x=180\).
Combine like terms: \((10x-2x)+(28 + 52)=180\), which gives \(8x+80 = 180\).
Subtract 80 from both sides: \(8x=180 - 80=100\). Then \(x=\frac{100}{8}=\frac{25}{2}\).

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

Substitute \(x = 2\) into the expression for \(\angle MPQ\). \(\angle MPQ=(10x + 28)^{\circ}\).
When \(x = 10\) (correction: from \(8x=100\), \(x = 10\)), \(\angle MPQ=10\times10+28=128^{\circ}\).

Answer:

\(128\)