Question

1. if \\( \overline { n p } \\) bisects \\( \angle m n q, m \angle m n q = ( 8 x + 12 ) ^ { \circ }, m \angle p n q = 78 ^ { \circ } \\), and \\( m \angle r n m = ( 3 y - 9 ) ^ { \circ } \\), find the values of \\( x \\) and \\( y \\).

Explanation:

Step1: Find the value of \(x\)

Since \(\overline{NP}\) bisects \(\angle MNQ\), then \(m\angle MNQ = 2m\angle PNQ\).
Given \(m\angle MNQ=(8x + 12)^{\circ}\) and \(m\angle PNQ = 78^{\circ}\), we have the equation \(8x+12=2\times78\).
Simplify the right - hand side: \(8x+12 = 156\).
Subtract 12 from both sides: \(8x=156 - 12\), so \(8x=144\).
Divide both sides by 8: \(x=\frac{144}{8}=18\).

Step2: Find the value of \(y\)

\(\angle RNM\) and \(\angle MNQ\) are vertical angles. Vertical angles are equal.
Since \(m\angle MNQ = 156^{\circ}\) (from step 1, \(8x + 12\) with \(x = 18\): \(8\times18+12=144 + 12=156\)), and \(m\angle RNM=(3y - 9)^{\circ}\).
Set up the equation \(3y-9=156\).
Add 9 to both sides: \(3y=156 + 9\), so \(3y=165\).
Divide both sides by 3: \(y=\frac{165}{3}=55\).

Answer:

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