Parallel Lines cut by a transversal | fjazz | Live - Free Printable

Problem Analysis:

The image shows two parallel lines \( y \) and \( z \) cut by a transversal. We are tasked with determining the measures of all missing angles and identifying the relationships between specific angle pairs.

#### Key Concepts:
1. Corresponding Angles: Angles that are in the same relative position at each intersection. They are congruent.
2. Alternate Interior Angles: Angles on opposite sides of the transversal and inside the parallel lines. They are congruent.
3. Alternate Exterior Angles: Angles on opposite sides of the transversal and outside the parallel lines. They are congruent.
4. Consecutive Interior Angles (Same-Side Interior Angles): Angles on the same side of the transversal and inside the parallel lines. They are supplementary (sum to 180°).
5. Vertical Angles: Angles opposite each other when two lines intersect. They are congruent.
6. Supplementary Angles: Angles that sum to 180°.

Step-by-Step Solution:

#### Given:
- \( \angle b = 74^\circ \)
- \( \angle 13 = 39^\circ \)

#### Determine Missing Angles:

1. Top Section (Above Line \( y \)):
- \( \angle 1 = 74^\circ \) (corresponding to \( \angle b \))
- \( \angle 2 = 180^\circ - 74^\circ = 106^\circ \) (supplementary to \( \angle 1 \))
- \( \angle 3 = 74^\circ \) (vertical to \( \angle 1 \))
- \( \angle 4 = 106^\circ \) (corresponding to \( \angle 2 \))
- \( \angle 5 = 74^\circ \) (alternate exterior to \( \angle 1 \))
- \( \angle 6 = 106^\circ \) (alternate interior to \( \angle 2 \))
- \( \angle 7 = 74^\circ \) (vertical to \( \angle 5 \))

2. Bottom Section (Below Line \( z \)):
- \( \angle 13 = 39^\circ \) (given)
- \( \angle 14 = 180^\circ - 39^\circ = 141^\circ \) (supplementary to \( \angle 13 \))
- \( \angle 15 = 39^\circ \) (vertical to \( \angle 13 \))
- \( \angle 16 = 141^\circ \) (corresponding to \( \angle 14 \))
- \( \angle 17 = 39^\circ \) (alternate exterior to \( \angle 13 \))
- \( \angle 18 = 141^\circ \) (alternate interior to \( \angle 14 \))
- \( \angle 12 = 39^\circ \) (vertical to \( \angle 17 \))

3. Middle Section (Between Lines \( y \) and \( z \)):
- \( \angle 8 = 141^\circ \) (corresponding to \( \angle 14 \))
- \( \angle 9 = 39^\circ \) (alternate interior to \( \angle 13 \))
- \( \angle 10 = 141^\circ \) (alternate exterior to \( \angle 14 \))
- \( \angle 11 = 39^\circ \) (vertical to \( \angle 9 \))

#### Final Angle Measures:
\[
\begin{aligned}
&\angle 1 = 74^\circ, \quad \angle 2 = 106^\circ, \quad \angle 3 = 74^\circ, \quad \angle 4 = 106^\circ, \\
&\angle 5 = 74^\circ, \quad \angle 6 = 106^\circ, \quad \angle 7 = 74^\circ, \\
&\angle 8 = 141^\circ, \quad \angle 9 = 39^\circ, \quad \angle 10 = 141^\circ, \quad \angle 11 = 39^\circ, \\
&\angle 12 = 39^\circ, \quad \angle 13 = 39^\circ, \quad \angle 14 = 141^\circ, \quad \angle 15 = 39^\circ, \\
&\angle 16 = 141^\circ, \quad \angle 17 = 39^\circ, \quad \angle 18 = 141^\circ.
\end{aligned}
\]

#### Name the Relationships Between Angle Pairs:
1. \( \angle 9 \) & \( \angle 11 \):
- Both are vertical angles. Relationship: Vertical Angles

2. \( \angle 15 \) & \( \angle 3 \):
- Both are alternate interior angles. Relationship: Alternate Interior Angles

3. \( \angle 14 \) & \( \angle 5 \):
- Both are corresponding angles. Relationship: Corresponding Angles

Final Answer:

\[
\boxed{
\begin{aligned}
&\angle 1 = 74^\circ, \angle 2 = 106^\circ, \angle 3 = 74^\circ, \angle 4 = 106^\circ, \angle 5 = 74^\circ, \angle 6 = 106^\circ, \angle 7 = 74^\circ, \\
&\angle 8 = 141^\circ, \angle 9 = 39^\circ, \angle 10 = 141^\circ, \angle 11 = 39^\circ, \angle 12 = 39^\circ, \angle 13 = 39^\circ, \angle 14 = 141^\circ, \\
&\angle 15 = 39^\circ, \angle 16 = 141^\circ, \angle 17 = 39^\circ, \angle 18 = 141^\circ. \\
&\text{Relationships:} \\
&\text{a. } \angle 9 \text{ & } \angle 11: \text{Vertical Angles} \\
&\text{b. } \angle 15 \text{ & } \angle 3: \text{Alternate Interior Angles} \\
&\text{c. } \angle 14 \text{ & } \angle 5: \text{Corresponding Angles}
\end{aligned}
}
\]