Parallel Lines cut by a Transversal Practice | StudyX - Free Printable

Problem: Solve for \( x \) and \( y \) in each of the given diagrams involving parallel lines cut by a transversal.

We will use the properties of angles formed when parallel lines are cut by a transversal:
1. Corresponding Angles: Angles in matching corners are equal.
2. Alternate Interior Angles: Angles on opposite sides of the transversal and inside the parallel lines are equal.
3. Alternate Exterior Angles: Angles on opposite sides of the transversal and outside the parallel lines are equal.
4. Consecutive Interior Angles (Same-Side Interior Angles): Angles on the same side of the transversal and inside the parallel lines are supplementary (sum to 180°).
5. Vertical Angles: Opposite angles formed by intersecting lines are equal.
6. Supplementary Angles: Angles that sum to 180°.

Let's solve each problem step by step.

---

Problem 1:


#### Given:
- One angle is \( 45^\circ \).
- Another angle is labeled \( x^\circ \).
- A third angle is labeled \( y^\circ \).

#### Solution:
1. The angle \( x^\circ \) is a vertical angle to the given \( 45^\circ \) angle. Vertical angles are equal:
\[
x = 45^\circ
\]

2. The angle \( y^\circ \) is a consecutive interior angle to the given \( 45^\circ \) angle. Consecutive interior angles are supplementary:
\[
y + 45^\circ = 180^\circ
\]
Solving for \( y \):
\[
y = 180^\circ - 45^\circ = 135^\circ
\]

#### Final Answer for Problem 1:
\[
\boxed{x = 45^\circ, y = 135^\circ}
\]

---

Problem 2:


#### Given:
- One angle is \( 60^\circ \).
- Another angle is labeled \( x^\circ \).
- A third angle is labeled \( y^\circ \).

#### Solution:
1. The angle \( x^\circ \) is a corresponding angle to the given \( 60^\circ \) angle. Corresponding angles are equal:
\[
x = 60^\circ
\]

2. The angle \( y^\circ \) is a consecutive interior angle to the given \( 60^\circ \) angle. Consecutive interior angles are supplementary:
\[
y + 60^\circ = 180^\circ
\]
Solving for \( y \):
\[
y = 180^\circ - 60^\circ = 120^\circ
\]

#### Final Answer for Problem 2:
\[
\boxed{x = 60^\circ, y = 120^\circ}
\]

---

Problem 3:


#### Given:
- One angle is \( 119^\circ \).
- Another angle is labeled \( x^\circ \).
- A third angle is labeled \( y^\circ \).

#### Solution:
1. The angle \( x^\circ \) is an alternate interior angle to the given \( 119^\circ \) angle. Alternate interior angles are equal:
\[
x = 119^\circ
\]

2. The angle \( y^\circ \) is a consecutive interior angle to the given \( 119^\circ \) angle. Consecutive interior angles are supplementary:
\[
y + 119^\circ = 180^\circ
\]
Solving for \( y \):
\[
y = 180^\circ - 119^\circ = 61^\circ
\]

#### Final Answer for Problem 3:
\[
\boxed{x = 119^\circ, y = 61^\circ}
\]

---

Problem 4:


#### Given:
- One angle is \( 124^\circ \).
- Another angle is labeled \( x^\circ \).

#### Solution:
1. The angle \( x^\circ \) is an alternate exterior angle to the given \( 124^\circ \) angle. Alternate exterior angles are equal:
\[
x = 124^\circ
\]

#### Final Answer for Problem 4:
\[
\boxed{x = 124^\circ}
\]

---

Problem 5:


#### Given:
- One angle is \( 43^\circ \).
- Another angle is labeled \( x^\circ \).
- A third angle is labeled \( y^\circ \).

#### Solution:
1. The angle \( x^\circ \) is a corresponding angle to the given \( 43^\circ \) angle. Corresponding angles are equal:
\[
x = 43^\circ
\]

2. The angle \( y^\circ \) is a vertical angle to the given \( 43^\circ \) angle. Vertical angles are equal:
\[
y = 43^\circ
\]

#### Final Answer for Problem 5:
\[
\boxed{x = 43^\circ, y = 43^\circ}
\]

---

Problem 6:


#### Given:
- One angle is \( 151^\circ \).
- Another angle is labeled \( x^\circ \).
- A third angle is labeled \( y^\circ \).

#### Solution:
1. The angle \( x^\circ \) is a consecutive interior angle to the given \( 151^\circ \) angle. Consecutive interior angles are supplementary:
\[
x + 151^\circ = 180^\circ
\]
Solving for \( x \):
\[
x = 180^\circ - 151^\circ = 29^\circ
\]

2. The angle \( y^\circ \) is a corresponding angle to the angle \( x^\circ \). Corresponding angles are equal:
\[
y = x = 29^\circ
\]

#### Final Answer for Problem 6:
\[
\boxed{x = 29^\circ, y = 29^\circ}
\]

---

Final Answers:

1. \( \boxed{x = 45^\circ, y = 135^\circ} \)
2. \( \boxed{x = 60^\circ, y = 120^\circ} \)
3. \( \boxed{x = 119^\circ, y = 61^\circ} \)
4. \( \boxed{x = 124^\circ} \)
5. \( \boxed{x = 43^\circ, y = 43^\circ} \)
6. \( \boxed{x = 29^\circ, y = 29^\circ} \)