Intersecting Lines and Angles worksheet with six problems requiring calculation of angle measures.

To solve the problems involving intersecting lines and angles, we need to use key geometric properties related to angles formed by intersecting lines. Here are the steps and solutions for each problem:

---

Key Concepts:

1. Vertical Angles: When two lines intersect, the angles opposite each other (vertical angles) are congruent.
- If \( \angle 1 = x \), then \( \angle 3 = x \).
- If \( \angle 2 = y \), then \( \angle 4 = y \).

2. Supplementary Angles: Angles that form a linear pair (adjacent angles on a straight line) are supplementary, meaning their measures add up to \( 180^\circ \).
- If \( \angle 1 \) and \( \angle 2 \) are adjacent and form a straight line, then \( \angle 1 + \angle 2 = 180^\circ \).

---

Problem 1:

Given:
- \( \angle 1 = 135^\circ \)

Using the properties of vertical angles:
- \( \angle 3 = \angle 1 = 135^\circ \)

Using the property of supplementary angles:
- \( \angle 1 + \angle 2 = 180^\circ \)
\[
135^\circ + \angle 2 = 180^\circ \implies \angle 2 = 45^\circ
\]
- Similarly, \( \angle 4 = \angle 2 = 45^\circ \)

Solution:
\[
\angle 2 = 45^\circ, \quad \angle 3 = 135^\circ, \quad \angle 4 = 45^\circ
\]

---

Problem 2:

Given:
- \( \angle 1 = 63^\circ \)

Using the properties of vertical angles:
- \( \angle 3 = \angle 1 = 63^\circ \)

Using the property of supplementary angles:
- \( \angle 1 + \angle 2 = 180^\circ \)
\[
63^\circ + \angle 2 = 180^\circ \implies \angle 2 = 117^\circ
\]
- Similarly, \( \angle 4 = \angle 2 = 117^\circ \)

Solution:
\[
\angle 2 = 117^\circ, \quad \angle 3 = 63^\circ, \quad \angle 4 = 117^\circ
\]

---

Problem 3:

Given:
- \( \angle 1 = 119^\circ \)

Using the properties of vertical angles:
- \( \angle 3 = \angle 1 = 119^\circ \)

Using the property of supplementary angles:
- \( \angle 1 + \angle 2 = 180^\circ \)
\[
119^\circ + \angle 2 = 180^\circ \implies \angle 2 = 61^\circ
\]
- Similarly, \( \angle 4 = \angle 2 = 61^\circ \)

Solution:
\[
\angle 2 = 61^\circ, \quad \angle 3 = 119^\circ, \quad \angle 4 = 61^\circ
\]

---

Problem 4:

Given:
- \( \angle 2 = 25^\circ \)

Using the properties of vertical angles:
- \( \angle 4 = \angle 2 = 25^\circ \)

Using the property of supplementary angles:
- \( \angle 1 + \angle 2 = 180^\circ \)
\[
\angle 1 + 25^\circ = 180^\circ \implies \angle 1 = 155^\circ
\]
- Similarly, \( \angle 3 = \angle 1 = 155^\circ \)

Solution:
\[
\angle 1 = 155^\circ, \quad \angle 3 = 155^\circ, \quad \angle 4 = 25^\circ
\]

---

Problem 5:

Given:
- \( \angle 1 = 137^\circ \)

Using the properties of vertical angles:
- \( \angle 3 = \angle 1 = 137^\circ \)

Using the property of supplementary angles:
- \( \angle 1 + \angle 2 = 180^\circ \)
\[
137^\circ + \angle 2 = 180^\circ \implies \angle 2 = 43^\circ
\]
- Similarly, \( \angle 4 = \angle 2 = 43^\circ \)

Solution:
\[
\angle 2 = 43^\circ, \quad \angle 3 = 137^\circ, \quad \angle 4 = 43^\circ
\]

---

Problem 6:

Given:
- \( \angle 2 = 69^\circ \)

Using the properties of vertical angles:
- \( \angle 4 = \angle 2 = 69^\circ \)

Using the property of supplementary angles:
- \( \angle 1 + \angle 2 = 180^\circ \)
\[
\angle 1 + 69^\circ = 180^\circ \implies \angle 1 = 111^\circ
\]
- Similarly, \( \angle 3 = \angle 1 = 111^\circ \)

Solution:
\[
\angle 1 = 111^\circ, \quad \angle 3 = 111^\circ, \quad \angle 4 = 69^\circ
\]

---

Final Answers:

\[
\boxed{
\begin{array}{ll}
\text{Problem 1:} & \angle 2 = 45^\circ, \angle 3 = 135^\circ, \angle 4 = 45^\circ \\
\text{Problem 2:} & \angle 2 = 117^\circ, \angle 3 = 63^\circ, \angle 4 = 117^\circ \\
\text{Problem 3:} & \angle 2 = 61^\circ, \angle 3 = 119^\circ, \angle 4 = 61^\circ \\
\text{Problem 4:} & \angle 1 = 155^\circ, \angle 3 = 155^\circ, \angle 4 = 25^\circ \\
\text{Problem 5:} & \angle 2 = 43^\circ, \angle 3 = 137^\circ, \angle 4 = 43^\circ \\
\text{Problem 6:} & \angle 1 = 111^\circ, \angle 3 = 111^\circ, \angle 4 = 69^\circ \\
\end{array}
}
\]