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The diagram shows two parallel roads \(a\) and \(b\), and several angles formed by intersections at points \(P\), \(Q\), \(R\), \(S\), \(T\), \(U\), and \(V\). You are tasked with solving for \(x\) and finding the measure of \(\angle VTU\).

### Step-by-step approach:

1. **Identify relationships between angles:**
   Since roads \(a\) and \(b\) are parallel, corresponding angles, alternate interior angles, and vertically opposite angles can be used to relate the given angle expressions.

2. **Set up an equation using vertical angles:**
   From the diagram:
   - \(\angle PQR = (x + 40)^\circ\)
   - \(\angle RVS = (2x - 22)^\circ\)

   These two angles are vertically opposite and must be equal:
   \[
   x + 40 = 2x - 22
   \]

3. **Solve for \(x\):**
   Rearrange the equation:
   \[
   40 + 22 = 2x - x
   \]
   \[
   62 = x
   \]

4. **Find the measure of \(\angle VTU\):**
   Using the value \(x = 62\), substitute into the expression for \(\angle RVS\):
   \[
   \angle RVS = 2(62) - 22 = 124 - 22 = 102^\circ
   \]
   Since \(\angle VTU\) corresponds to \(\angle RVS\), the measure of \(\angle VTU\) is \(102^\circ\).

### Final answers:
- \(x = 62\)
- \(\text{m}\angle VTU = 102^\circ\)

Would you like more details or have any questions?

Here are five related questions for further practice:
1. What are corresponding angles, and how can they help solve parallel line problems?
2. How do you identify alternate interior angles in a parallel line diagram?
3. What other angle pairs (like supplementary or complementary) are useful in geometric proofs?
4. How do vertically opposite angles relate in terms of equality?
5. If the lines \(a\) and \(b\) were not parallel, how would this affect the solution?

**Tip:** When working with parallel lines, always look for corresponding or alternate angles first to set up relationships between variables.

In the diagram, roads a and b are parallel. Solve for x and find the measure of ∠VTU.

Learn how to solve for x and find the measure of ∠VTU in a geometry problem involving parallel lines and vertical angles. This step-by-step solution is suitable for Grades 7-10 students.

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