Question

1. if (overline{np}) bisects (angle mnq), (mangle mnq=(8x + 12)^{circ}), (mangle pnq = 78^{circ}), and (mangle rnm=(3y - 9)^{circ}), find the values of (x) and (y).

Explanation:

Step1: Use the angle - bisector property

Since \( \overline{NP}\) bisects \( \angle MNQ\), then \(m\angle PNO=m\angle PNQ = 78^{\circ}\). Also, \(m\angle MNQ=2\times m\angle PNQ\). Given \(m\angle MNQ=(8x + 12)^{\circ}\), we have the equation \(8x+12 = 2\times78\).

Step2: Solve for \(x\)

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Step3: Use the vertical - angle property

\(\angle RNM\) and \( \angle MNQ\) are vertical angles. So \(m\angle RNM=m\angle MNQ\). Given \(m\angle RNM=(3y - 9)^{\circ}\) and \(m\angle MNQ = 156^{\circ}\), we have the equation \(3y-9=156\).

Step4: Solve for \(y\)

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Answer:

\(x = 18\) and \(y = 55\)