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The diagram shows two intersecting lines with point \( Q \) as the intersection. The angle \( \angle PQR \) is labeled \( (3x - 5)^\circ \), and the adjacent angle \( \angle RQS \) is labeled \( (x + 1)^\circ \).

These two angles are **supplementary** because they form a straight line, meaning their measures must add up to 180 degrees. Therefore, we can set up the following equation:

\[
(3x - 5)^\circ + (x + 1)^\circ = 180^\circ
\]

Let's solve for \( x \):

\[
3x - 5 + x + 1 = 180
\]

\[
4x - 4 = 180
\]

\[
4x = 184
\]

\[
x = 46
\]

Now that we know \( x = 46 \), we can substitute this value back into the expression for \( m \angle PQR \):

\[
m \angle PQR = 3x - 5 = 3(46) - 5 = 138 - 5 = 133^\circ
\]

Thus, the measure of \( \angle PQR \) is \( 133^\circ \).

Would you like more details or have any questions?

Here are 5 related questions you might explore:
1. How do we know that \( \angle PQR \) and \( \angle RQS \) are supplementary?
2. What would happen if these angles were not supplementary?
3. Can we find the measure of \( \angle RQS \) using the value of \( x \)?
4. How would the solution change if the angles formed a triangle instead of a straight line?
5. What is the general rule for supplementary angles?

**Tip:** Always check if the angles form a straight line or other relationships like vertical angles or angles in a triangle to set up your equation.

Learn how to find the measure of angle PQR by solving a system of equations involving supplementary angles. This problem includes algebraic expressions and key geometry concepts suitable for Grades 8-10 students.

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