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The problem involves a pair of parallel lines and some angles, and we are tasked with finding angles \( x \) and \( y \). Here’s the approach to solve it:

1. **Identify Relationships Using Parallel Lines:**
   - Given that \( PQ \parallel SR \) and \( PT \parallel NM \), we can apply the concepts of alternate interior angles and corresponding angles.

2. **Angle \( x \):**
   - \( \angle PTS = 55^\circ \) (given). 
   - Since \( PT \parallel NM \), \( \angle PTS = \angle QNM \) by the alternate interior angle theorem.
   - Therefore, \( x = 55^\circ \).

3. **Angle \( y \):**
   - Now, let’s focus on the angles at point \( T \). 
   - Since \( PQ \parallel SR \), we can use the concept of corresponding angles. \( \angle RTM = 70^\circ \) (given), and \( \angle RTM \) and \( \angle PTM \) form a linear pair.
   - The sum of a linear pair is \( 180^\circ \), so \( y = 180^\circ - 70^\circ = 110^\circ \).

Thus, the values of the angles are:
- \( x = 55^\circ \)
- \( y = 110^\circ \)

Would you like further details or have any other questions?

Here are five related questions for deeper understanding:
1. What are alternate interior angles, and how do they relate to parallel lines?
2. Can corresponding angles ever be unequal if the lines are parallel?
3. How does the linear pair relationship help in finding unknown angles?
4. Could \( x \) and \( y \) change if \( PT \) and \( NM \) were not parallel?
5. How do we identify when angles form a linear pair in more complex diagrams?

**Tip:** Always look for parallel lines in geometry problems—they often simplify angle calculations through properties like corresponding and alternate interior angles.

In the figure given, PQ || SR and PT || NM. Find ∠x and ∠y.

Learn how to solve for angles ∠x and ∠y in a geometry problem involving parallel lines and interior angles. This detailed solution covers key geometric concepts such as the alternate interior angles theorem and the corresponding angles theorem. Suitable for students in Grades 8-10.

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