Diagrams illustrating angle relationships formed by a transversal intersecting two parallel lines, with variables representing unknown angles.

To solve the problems in the image, we need to use properties of angles formed by a transversal intersecting parallel lines. Specifically, we will use the following key concepts:

1. Corresponding Angles: When two parallel lines are cut by a transversal, corresponding angles are equal.
2. Alternate Interior Angles: When two parallel lines are cut by a transversal, alternate interior angles are equal.
3. Same-Side Interior Angles: When two parallel lines are cut by a transversal, same-side interior angles are supplementary (sum to 180°).
4. Vertical Angles: Vertical angles are always equal.

Let's solve each problem step by step.

---

Problem 1:

[](https://i.imgur.com/1234567.png)

#### Given:
- \( l \parallel m \)
- One angle is \( 110^\circ \)
- Another angle is \( x^\circ \)

#### Solution:
The angle \( x^\circ \) is a corresponding angle to the given \( 110^\circ \) angle because they are on the same side of the transversal and above the parallel lines. Since corresponding angles are equal when the lines are parallel:
\[
x = 110^\circ
\]

#### Answer:
\[
\boxed{110}
\]

---

Problem 2:

[](https://i.imgur.com/2345678.png)

#### Given:
- \( l \parallel m \)
- One angle is \( 84^\circ \)
- Another angle is \( y^\circ \)

#### Solution:
The angle \( y^\circ \) is an alternate interior angle to the given \( 84^\circ \) angle because they are on opposite sides of the transversal and between the parallel lines. Since alternate interior angles are equal when the lines are parallel:
\[
y = 84^\circ
\]

#### Answer:
\[
\boxed{84}
\]

---

Problem 3:

[](https://i.imgur.com/3456789.png)

#### Given:
- \( l \parallel m \)
- One angle is \( 100^\circ \)
- Another angle is \( z^\circ \)

#### Solution:
The angle \( z^\circ \) is a same-side interior angle to the given \( 100^\circ \) angle because they are on the same side of the transversal and between the parallel lines. Since same-side interior angles are supplementary when the lines are parallel:
\[
z + 100^\circ = 180^\circ
\]
Solving for \( z \):
\[
z = 180^\circ - 100^\circ = 80^\circ
\]

#### Answer:
\[
\boxed{80}
\]

---

Problem 4:

[](https://i.imgur.com/4567890.png)

#### Given:
- \( l \parallel m \)
- One angle is \( 53^\circ \)
- Another angle is \( s^\circ \)

#### Solution:
The angle \( s^\circ \) is a vertical angle to the given \( 53^\circ \) angle because they are opposite each other at the intersection of the transversal. Since vertical angles are always equal:
\[
s = 53^\circ
\]

#### Answer:
\[
\boxed{53}
\]

---

Final Answers:

1. \(\boxed{110}\)
2. \(\boxed{84}\)
3. \(\boxed{80}\)
4. \(\boxed{53}\)