https://file.aiquickdraw.com/mathbot/file/17259694078959urp17iq.jpg

Let's analyze the problem from the image.

The figure consists of several intersecting lines with angles marked as expressions involving variables \( x \) and \( y \).

### Step-by-Step Solution:

#### Part (a) Determine the value of \( x \).

1. **Identifying Angle Relationships:**

   - The angles \( (2x + 25)^\circ \) and \( (3x - 5)^\circ \) are vertically opposite angles. Vertically opposite angles are always equal.
   - Thus, we have the equation:
   \[
   2x + 25 = 3x - 5
   \]

2. **Solving for \( x \):**

   - Rearrange the equation to isolate \( x \):
   \[
   25 + 5 = 3x - 2x \implies 30 = x
   \]
   - Therefore, the value of \( x \) is:
   \[
   x = 30
   \]

#### Part (b) Determine the value of \( y \).

1. **Identifying Angle Relationships:**

   - The angle \( (2y + 40)^\circ \) and the angle \( (3x - 5)^\circ \) are supplementary because they form a straight line (adjacent angles on a straight line add up to 180°).
   - Since we found \( x = 30 \), substitute to find the value of the angle \( (3x - 5) \):
   \[
   3(30) - 5 = 90 - 5 = 85^\circ
   \]

   - Therefore, the equation for supplementary angles is:
   \[
   (2y + 40) + 85 = 180
   \]

2. **Solving for \( y \):**

   - Simplify the equation:
   \[
   2y + 125 = 180
   \]
   \[
   2y = 180 - 125
   \]
   \[
   2y = 55
   \]
   \[
   y = 27.5
   \]

### Final Answer:

- The value of \( x \) is **30**.
- The value of \( y \) is **27.5**.

### Reasoning:

- The value of \( x \) was determined using the property that vertically opposite angles are equal.
- The value of \( y \) was determined using the property that adjacent angles on a straight line are supplementary (sum to 180°).

Would you like more details or have any questions?

### Related Questions:

1. What are vertically opposite angles, and why are they always equal?
2. Can we determine the angles \( \angle USP \) and \( \angle QPT \) using the values of \( x \) and \( y \)?
3. How can we find other unknown angles in the diagram?
4. What is the significance of supplementary angles, and where are they commonly used in geometry?
5. If we change the value of \( x \), how will it affect the value of \( y \)?

### Tip:

When solving problems involving intersecting lines, always look for vertically opposite angles or supplementary angles, as these relationships are key to finding unknown values.

Determine the values of x and y based on the given angles using algebra. The angles are represented by expressions: 2x + 25°, 3x - 5°, and 2y + 40°, with vertically opposite and supplementary angle relationships.

Learn how to determine the values of x and y using algebra. This problem explores vertically opposite angles and supplementary angles. Suitable for students in Grades 8-10.

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