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To solve for \( m \angle QPR \), let's use the information given:

### Given:
1. \( m \angle QPS = 107^\circ \)
2. \( m \angle RPS = 4x + 27^\circ \)
3. \( m \angle QPR = 9x - 115^\circ \)

We know that the sum of the angles around point \( P \) should add up to \( 180^\circ \) because the points form a straight line.

\[
m \angle QPS = m \angle QPR + m \angle RPS
\]

Substitute the given values:

\[
107^\circ = (9x - 115^\circ) + (4x + 27^\circ)
\]

Simplify the equation:

\[
107^\circ = 9x - 115^\circ + 4x + 27^\circ
\]
\[
107^\circ = 13x - 88^\circ
\]
Add \( 88^\circ \) to both sides:

\[
107^\circ + 88^\circ = 13x
\]
\[
195^\circ = 13x
\]
Now, solve for \( x \):

\[
x = \frac{195^\circ}{13} = 15^\circ
\]

### Find \( m \angle QPR \):

Substitute \( x = 15^\circ \) into the expression for \( m \angle QPR \):

\[
m \angle QPR = 9x - 115^\circ = 9(15^\circ) - 115^\circ = 135^\circ - 115^\circ = 20^\circ
\]

Thus, the measure of \( \angle QPR \) is \( 20^\circ \).

Let me know if you would like further clarification or more details.

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Here are 5 related questions:
1. How would you verify that the sum of the angles around point \( P \) adds up to \( 180^\circ \)?
2. What is the value of \( m \angle RPS \) once \( x \) is known?
3. Can this method be applied if the angles were not on a straight line? Why or why not?
4. How does the value of \( x \) affect the individual angles?
5. Could there be an alternative method to find \( m \angle QPR \)?

**Tip:** Always double-check your algebraic manipulation when solving multi-variable angle equations, as small errors can significantly impact the result.

Dadas las relaciones: m∠QPS = 107°, m∠RPS = 4x + 27°, m∠QPR = 9x - 115°. Encuentra m∠QPR.

Learn how to solve for the angle m∠QPR using the linear pair of angles concept. The sum of angles around point P adds up to 180°, including expressions involving variables. Ideal for students in Grades 8-10.

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