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### Problem 31
**Given:** Lines \( l \parallel m \). Find \( x \).

In the triangle \( \triangle ABC \):
- The sum of angles in a triangle is 180°.
- Given angles:
  - \( \angle B = 50^\circ \)
  - \( \angle C = 115^\circ \)
  
Using the angle sum property:
\[
x + 50^\circ + 115^\circ = 180^\circ
\]
\[
x + 165^\circ = 180^\circ
\]
\[
x = 15^\circ
\]

**Answer:** \( x = 15^\circ \)

---

### Problem 32
**Given:** \( l \parallel m \) and \( n \) is a transversal. Find \( x \).

Since \( l \parallel m \), corresponding angles are equal.
1. \( 3y = x + 15^\circ \)
2. \( 2y + 25^\circ = 3y \)

Solving the second equation:
\[
2y + 25^\circ = 3y
\]
\[
25^\circ = y
\]

Substitute \( y = 25^\circ \) into the first equation:
\[
3(25^\circ) = x + 15^\circ
\]
\[
75^\circ = x + 15^\circ
\]
\[
x = 60^\circ
\]

**Answer:** \( x = 60^\circ \), \( y = 25^\circ \)

---

### Problem 33
**For what value of \( x \), \( l \parallel m \).**

Given angles in the figure:
- \( 50^\circ \) (corresponding angle to \( x - 20^\circ \))
- \( 120^\circ \) (alternate angle)

Since \( l \parallel m \), the sum of the interior angles on the same side of the transversal should be 180°:
\[
50^\circ + (x - 20^\circ) = 120^\circ
\]
\[
x = 90^\circ
\]

---

### Problem 

Find the values of x and y for the given geometry problems involving parallel lines and transversals.

Learn how to find unknown angles in problems involving parallel lines and transversals. This guide includes step-by-step solutions for using the angle sum property and theorems on corresponding and alternate interior angles. Suitable for Grades 8-10.

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