The problem appears to involve solving for $x$ in a geometric configuration, with angle measures expressed algebraically. Let me analyze the image and extract the relevant information to provide a solution.

Problem Breakdown:

• Given:

  • $\overline{PS}$ bisects $\angle PQK$.
  • The measure of $\angle PQK$ is represented as $4x - 20$.
  • The measure of $\angle PQR$ is $42^\circ$.

• Goal:

  • Solve for $x$.

Since $\overline{PS}$ bisects $\angle PQK$, it implies that $\angle PQS = \angle SQK$. This symmetry will be useful for setting up an equation. Let me calculate the solution for $x$.

I'll write out the steps.The value of $x$ that satisfies the given conditions is $\frac{79}{2}$, or $39.5$.

Explanation:

• Since $\overline{PS}$ bisects $\angle PQK$, the two smaller angles formed by the bisection are equal, each being $\frac{4x - 20}{2}$.
• Using the triangle sum property, the sum of the angles in triangle $\triangle PQR$ must be $180^\circ$.
• By solving for $x$, we found $x = 39.5$.

Would you like further details or have any additional questions?

Here are some related questions to explore:

1. What happens if the bisector condition changes to another angle?
2. How does the triangle sum property apply in different types of triangles?
3. What is the effect of changing angle $\angle PQR$ on the solution?
4. Can the same method be applied if the bisector is external to the triangle?
5. How would you solve this problem using a geometric construction?

Tip: When solving geometry problems, visualizing the relationships and symmetries in the diagram can simplify algebraic expressions and make problem-solving easier.