https://file.aiquickdraw.com/mathbot/file/1726187452266bdw5ppp6.jpg

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 \(180^\circ\).

∠RQT is a straight angle. What are m∠RQS and m∠TQS?

Learn how to solve for m∠RQS and m∠TQS, given that ∠RQT is a straight angle. Using algebraic expressions and the sum of angles on a straight line, this step-by-step solution simplifies the problem for Grade 9 students.

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