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The problem is asking to find the measure of \( m \angle RPS \), given that:

- \( m \angle QPS \) is a straight angle, meaning it equals \( 180^\circ \).
- \( m \angle RPS = 6x + 11^\circ \).
- \( m \angle QPR = 7x + 143^\circ \).

Since \( \angle QPS \) is a straight angle, we know:

\[
m \angle QPS = m \angle QPR + m \angle RPS = 180^\circ
\]

Substitute the expressions for \( m \angle QPR \) and \( m \angle RPS \):

\[
(7x + 143^\circ) + (6x + 11^\circ) = 180^\circ
\]

Simplifying:

\[
13x + 154^\circ = 180^\circ
\]

Solve for \( x \):

\[
13x = 180^\circ - 154^\circ
\]
\[
13x = 26^\circ
\]
\[
x = \frac{26^\circ}{13} = 2^\circ
\]

Now that we know \( x = 2^\circ \), substitute this value back into the expression for \( m \angle RPS \):

\[
m \angle RPS = 6(2^\circ) + 11^\circ = 12^\circ + 11^\circ = 23^\circ
\]

Thus, the measure of \( m \angle RPS \) is \( 23^\circ \).

Let me know if you want further details or have any questions. 

Here are some related questions:
1. What are the properties of straight angles?
2. How would the problem change if \( m \angle QPS \) were a different type of angle?
3. What is the significance of solving for \( x \) in this context?
4. Can you apply the same method to other geometric figures?
5. How do linear pairs of angles relate to straight angles?

**Tip:** Always check your algebra when working with equations involving multiple variables!

Dadas las relaciones: m∠QPS es un ángulo llano. m∠RPS = 6x + 11°. m∠QPR = 7x + 143°. Encuentra m∠RPS.

Learn how to find the measure of angle RPS using algebraic expressions and geometry theorems. This problem demonstrates key concepts like solving for unknown variables in angles and the Angle Sum Theorem. Ideal for students in Grades 8-10.

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