Parallel Lines cut by a transversal worksheet - 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°.

Given Information:

- \( \angle b = 74^\circ \)
- \( \angle a = 39^\circ \)

Step-by-Step Solution:

#### Step 1: Identify and Calculate Angles Using Corresponding, Alternate, and Vertical Relationships

1. Top Section (Above Line \( y \)):
- \( \angle 1 \): Corresponds to \( \angle b \). Therefore, \( \angle 1 = 74^\circ \).
- \( \angle 2 \): Vertical to \( \angle 1 \). Therefore, \( \angle 2 = 74^\circ \).
- \( \angle 3 \): Corresponds to \( \angle a \). Therefore, \( \angle 3 = 39^\circ \).
- \( \angle 4 \): Vertical to \( \angle 3 \). Therefore, \( \angle 4 = 39^\circ \).
- \( \angle 5 \): Corresponds to \( \angle a \). Therefore, \( \angle 5 = 39^\circ \).
- \( \angle 6 \): Vertical to \( \angle 5 \). Therefore, \( \angle 6 = 39^\circ \).
- \( \angle 7 \): Corresponds to \( \angle b \). Therefore, \( \angle 7 = 74^\circ \).

2. Bottom Section (Below Line \( z \)):
- \( \angle 8 \): Corresponds to \( \angle b \). Therefore, \( \angle 8 = 74^\circ \).
- \( \angle 9 \): Vertical to \( \angle 8 \). Therefore, \( \angle 9 = 74^\circ \).
- \( \angle 10 \): Corresponds to \( \angle a \). Therefore, \( \angle 10 = 39^\circ \).
- \( \angle 11 \): Vertical to \( \angle 10 \). Therefore, \( \angle 11 = 39^\circ \).
- \( \angle 12 \): Corresponds to \( \angle b \). Therefore, \( \angle 12 = 74^\circ \).
- \( \angle 13 \): Vertical to \( \angle 12 \). Therefore, \( \angle 13 = 74^\circ \).
- \( \angle 14 \): Corresponds to \( \angle a \). Therefore, \( \angle 14 = 39^\circ \).
- \( \angle 15 \): Vertical to \( \angle 14 \). Therefore, \( \angle 15 = 39^\circ \).
- \( \angle 16 \): Corresponds to \( \angle b \). Therefore, \( \angle 16 = 74^\circ \).
- \( \angle 17 \): Vertical to \( \angle 16 \). Therefore, \( \angle 17 = 74^\circ \).
- \( \angle 18 \): Corresponds to \( \angle a \). Therefore, \( \angle 18 = 39^\circ \).

#### Step 2: Name the Relationships Between Specific Angle Pairs

a. \( \angle 9 \) & \( \angle 11 \):
- \( \angle 9 \) and \( \angle 11 \) are alternate interior angles because they are on opposite sides of the transversal and inside the parallel lines.

b. \( \angle 15 \) & \( \angle 3 \):
- \( \angle 15 \) and \( \angle 3 \) are corresponding angles because they are in the same relative position at their respective intersections.

c. \( \angle 14 \) & \( \angle 5 \):
- \( \angle 14 \) and \( \angle 5 \) are vertical angles because they are opposite each other at the intersection of the transversal and line \( z \).

Final Answer:

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