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We are given the following information from the problem:

1. \( \overrightarrow{NP} \) bisects \( \angle MNQ \).
2. \( m\angle MNQ = (8x + 12)^\circ \).
3. \( m\angle PNQ = 78^\circ \).
4. \( m\angle RNM = (3y - 9)^\circ \).

Since \( \overrightarrow{NP} \) bisects \( \angle MNQ \), we know that:

\[
m\angle MNQ = 2 \times m\angle PNQ
\]

This gives us:

\[
8x + 12 = 2 \times 78
\]

Simplifying this:

\[
8x + 12 = 156
\]

Now, we can solve for \( x \):

\[
8x = 156 - 12
\]
\[
8x = 144
\]
\[
x = \frac{144}{8} = 18
\]

Now, for \( y \), notice that:

Since \( \angle RNM \) is a straight angle, it must sum up to \( 180^\circ \) with \( \angle MNQ \). Therefore:

\[
m\angle RNM + m\angle MNQ = 180^\circ
\]

Substitute the values of \( m\angle RNM \) and \( m\angle MNQ \):

\[
(3y - 9) + (8x + 12) = 180
\]

We already know \( x = 18 \), so substitute this value into the equation:

\[
(3y - 9) + (8(18) + 12) = 180
\]

Simplify the equation:

\[
(3y - 9) + (144 + 12) = 180
\]
\[
3y - 9 + 156 = 180
\]
\[
3y + 147 = 180
\]
\[
3y = 180 - 147
\]
\[
3y = 33
\]
\[
y = \frac{33}{3} = 11
\]

Thus, the values of \( x \) and \( y \) are:

\[
x = 18 \quad \text{and} \quad y = 11
\]

Would you like a further breakdown of any step, or do you have any other questions? Here are some related questions you can try:

1. How do you calculate the bisected angle if one side is known?
2. What is the relationship between straight angles and adjacent angles?
3. How do you solve linear equations with two variables?
4. Can you derive the angles in a triangle given one angle and side lengths?
5. How does angle bisection affect geometric properties?

**Tip**: Always verify your equation setup, especially when dealing with angle bisectors, by ensuring that bisected angles correctly add up to the total.

If NP bisects ∠MNQ, m∠MNQ = (8x + 12)°, m∠PNQ = 78°, and m∠RNM = (3y - 9)°, find the values of x and y.

Find the values of x and y in the given geometric problem where NP bisects ∠MNQ, and angles are expressed in terms of x and y. This problem explores concepts such as angle bisectors, linear equations, and geometry fundamentals. Suitable for students in Grades 9-10.

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