In the image, we are given the following details:

• $\angle RQT$ is a straight angle, meaning that the sum of the angles $\angle RQS$ and $\angle TQS$ must equal $180^\circ$.
• We are given expressions for these angles:
  • $m\angle RQS = (9x + 1)^\circ$
  • $m\angle TQS = (5x + 11)^\circ$

Step 1: Set up the equation

Since $\angle RQT$ is a straight angle: $m\angle RQS + m\angle TQS = 180^\circ$ Substitute the given expressions: $(9x + 1) + (5x + 11) = 180$

Step 2: Simplify and solve for $x$

Combine like terms: $9x + 5x + 1 + 11 = 180$ $14x + 12 = 180$

Subtract 12 from both sides: $14x = 168$

Now divide by 14: $x = 12$

Step 3: Calculate the angles

Now substitute $x = 12$ into the expressions for $\angle RQS$ and $\angle TQS$.

For $m\angle RQS$: $m\angle RQS = 9x + 1 = 9(12) + 1 = 108 + 1 = 109^\circ$

For $m\angle TQS$: $m\angle TQS = 5x + 11 = 5(12) + 11 = 60 + 11 = 71^\circ$

Final Answer:

• $m\angle RQS = 109^\circ$
• $m\angle TQS = 71^\circ$

Would you like further clarification on any step?

Here are five related questions:

1. How can we check if two angles are supplementary?
2. What is the definition of a straight angle?
3. How do we combine like terms in algebraic expressions?
4. What are alternate ways to solve equations like this one?
5. Can angles be negative in geometry?

Tip: Always remember that the sum of angles forming a straight line is $180^\circ$.