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In this problem, we are given that lines \( r \) and \( s \) are parallel, and both are intersected by a transversal line \( t \). The figure shows an angle of \( 135^\circ \) between \( r \) and \( t \), and we are asked to find the measure of the angle \( x \).

### Solution:
When two parallel lines are intersected by a transversal, several angle relationships are created. One key property is that **alternate interior angles** are congruent (equal). Another relevant property is that **corresponding angles** are also equal.

1. The given angle is \( 135^\circ \).
2. Since lines \( r \) and \( s \) are parallel, the angle \( x \) is an alternate interior angle to the angle marked \( 135^\circ \).
3. Therefore, by the alternate interior angle theorem, \( x = 135^\circ \).

### Answer:
\[
x = 135^\circ
\]

Would you like more details on this explanation or have any other questions?

### Further Questions:
1. What are the other types of angles formed by a transversal intersecting two parallel lines?
2. How do you determine corresponding angles when two lines are intersected by a transversal?
3. What is the sum of interior angles on the same side of a transversal intersecting two parallel lines?
4. How would the angle measures change if the given angle was \( 45^\circ \) instead of \( 135^\circ \)?
5. How can you use these principles in real-world situations involving parallel lines and angles?

### Tip:
When working with parallel lines and a transversal, remember that alternate interior angles are always congruent, which can simplify finding unknown angles.

Na figura a seguir, as retas r e s são paralelas e interceptadas pela reta t. O ângulo x tem medida igual a?

Discover how to find the measure of angle x when two parallel lines are intersected by a transversal. This problem explores the alternate interior angle theorem, using a given 135° angle to determine the value of x. Ideal for students in Grades 8-10 studying geometry.

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