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.