https://file.aiquickdraw.com/mathbot/file/1720798253760zsgfyolv.png

Let's solve the problems in the given figures.

### Problem 1: Value of \(\frac{a+b}{x}\)

In the first figure, we have lines \( l_1 \parallel l_2 \). We are given several angles and need to find the value of \(\frac{a + b}{x}\).

1. From the figure:
    - \(\angle ABE = 2a\)
    - \(\angle EBC = b\)
    - \(\angle FDE = x\)
    - \(\angle BAD = a\)
    - \(\angle BDA = 2b\)

Since \( l_1 \parallel l_2 \), angles formed by a transversal are congruent or supplementary. Using the given angles:

- \(\angle ABE = 2a\)
- \(\angle EBC = b\)

Thus, \( \angle ABC = 2a + b\).

Next, consider triangle \( FDE \):

\[
\angle DEF = 90^\circ
\]

The exterior angle theorem states that an exterior angle is equal to the sum of the opposite interior angles.

\[
x = \angle ABE + \angle EBC = 2a + b
\]

Therefore,

\[
\frac{a + b}{x} = \frac{a + b}{2a + b}
\]

### Problem 2: Measure of Angles \(\angle DSC, \angle PAS, \angle SAB,\) and \(\angle CST\)

Given:
- \( SC \parallel AB \)
- \( AD \) is a ray
- \( RS \parallel PQ \)

**Angle Calculations:**

1. **\(\angle DSC\):**
    \[
    \angle DSC = 32^\circ \text{ (given directly)}
    \]

2. **\(\angle PAS\):**
    - Since \( RS \parallel PQ \) and \( \angle AST = 3x \), using corresponding angles:
    \[
    \angle PAS = 3x
    \]

3. **\(\angle SAB\):**
    - Using alternate interior angles:
    \[
    \angle SAB = 68^\circ \text{ (since } AB \parallel SC)
    \]

4. **\(\angle CST\):**
    - Using the parallel lines and the transversal:
    \[
    \angle CST = \angle PAS = 3x
    \]

Summarizing the solutions:

1. \(\frac{a + b}{x} = \frac{a + b}{2a + b}\)
2. \(\angle DSC = 32^\circ\)
3. \(\angle PAS = 3x\)
4. \(\angle SAB = 68^\circ\)
5. \(\angle CST = 3x\)

If you have any further questions or need more details, feel free to ask!

Here are some questions you might want to ask next:

1. How do we find the value of \(x\) given \(a\) and \(b\)?
2. What are the properties of corresponding angles?
3. Can you explain the exterior angle theorem in more detail?
4. How do parallel lines affect angle measures?
5. What other geometric properties can be used in these problems?

**Tip:** When dealing with parallel lines and transversals, always look for corresponding, alternate interior, and alternate exterior angles to help solve for unknowns.

Explore solutions to geometry problems involving parallel lines, angle properties, and triangle properties. Learn how to calculate angle measures and apply the exterior angle theorem in triangles. Suitable for high school students.

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