Student worksheet on parallel lines cut by a transversal, with answers filled in.

Problem Analysis and Solution

The worksheet involves finding the measures of angles formed when parallel lines are cut by a transversal. We will solve each problem step by step, using properties of parallel lines and transversals.

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#### Problem 1:
Given:
- \( m\angle 8 = 23^\circ \)
- The diagram shows two parallel lines cut by a transversal.

We need to find the measures of angles \( \angle 1, \angle 2, \angle 3, \angle 4, \angle 5, \angle 6, \) and \( \angle 7 \).

##### Reasoning:
1. Corresponding Angles: When a transversal intersects two parallel lines, corresponding angles are equal.
2. Alternate Interior Angles: Alternate interior angles are equal.
3. Same-Side Interior Angles: Same-side interior angles are supplementary (sum to \( 180^\circ \)).
4. Vertical Angles: Vertical angles are equal.

##### Step-by-Step Solution:

- Given: \( m\angle 8 = 23^\circ \).
- Find \( m\angle 1 \):
- \( \angle 1 \) and \( \angle 8 \) are corresponding angles.
- Therefore, \( m\angle 1 = m\angle 8 = 23^\circ \).

- Find \( m\angle 2 \):
- \( \angle 2 \) and \( \angle 8 \) are alternate interior angles.
- Therefore, \( m\angle 2 = m\angle 8 = 23^\circ \).

- Find \( m\angle 3 \):
- \( \angle 3 \) and \( \angle 8 \) are same-side interior angles.
- Therefore, \( m\angle 3 + m\angle 8 = 180^\circ \).
- \( m\angle 3 = 180^\circ - 23^\circ = 157^\circ \).

- Find \( m\angle 4 \):
- \( \angle 4 \) and \( \angle 8 \) are vertical angles.
- Therefore, \( m\angle 4 = m\angle 8 = 23^\circ \).

- Find \( m\angle 5 \):
- \( \angle 5 \) and \( \angle 8 \) are corresponding angles.
- Therefore, \( m\angle 5 = m\angle 8 = 23^\circ \).

- Find \( m\angle 6 \):
- \( \angle 6 \) and \( \angle 8 \) are alternate interior angles.
- Therefore, \( m\angle 6 = m\angle 8 = 23^\circ \).

- Find \( m\angle 7 \):
- \( \angle 7 \) and \( \angle 8 \) are same-side interior angles.
- Therefore, \( m\angle 7 + m\angle 8 = 180^\circ \).
- \( m\angle 7 = 180^\circ - 23^\circ = 157^\circ \).

##### Final Answers for Problem 1:
\[
\begin{aligned}
&\text{a. } m\angle 1 = 23^\circ \\
&\text{b. } m\angle 2 = 23^\circ \\
&\text{c. } m\angle 3 = 157^\circ \\
&\text{d. } m\angle 4 = 23^\circ \\
&\text{e. } m\angle 5 = 23^\circ \\
&\text{f. } m\angle 6 = 23^\circ \\
&\text{g. } m\angle 7 = 157^\circ \\
\end{aligned}
\]

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#### Problem 2:
Given:
- \( m\angle 9 = 97^\circ \)
- \( m\angle 12 = 114^\circ \)
- The diagram shows two parallel lines cut by a transversal.

We need to find the measures of angles \( \angle 1, \angle 2, \angle 3, \angle 4, \angle 5, \angle 6, \angle 7, \angle 8, \angle 13, \angle 14 \).

##### Reasoning:
1. Corresponding Angles: When a transversal intersects two parallel lines, corresponding angles are equal.
2. Alternate Interior Angles: Alternate interior angles are equal.
3. Same-Side Interior Angles: Same-side interior angles are supplementary (sum to \( 180^\circ \)).
4. Vertical Angles: Vertical angles are equal.

##### Step-by-Step Solution:

- Given: \( m\angle 9 = 97^\circ \) and \( m\angle 12 = 114^\circ \).

- Find \( m\angle 1 \):
- \( \angle 1 \) and \( \angle 9 \) are corresponding angles.
- Therefore, \( m\angle 1 = m\angle 9 = 97^\circ \).

- Find \( m\angle 2 \):
- \( \angle 2 \) and \( \angle 9 \) are same-side interior angles.
- Therefore, \( m\angle 2 + m\angle 9 = 180^\circ \).
- \( m\angle 2 = 180^\circ - 97^\circ = 83^\circ \).

- Find \( m\angle 3 \):
- \( \angle 3 \) and \( \angle 12 \) are corresponding angles.
- Therefore, \( m\angle 3 = m\angle 12 = 114^\circ \).

- Find \( m\angle 4 \):
- \( \angle 4 \) and \( \angle 12 \) are same-side interior angles.
- Therefore, \( m\angle 4 + m\angle 12 = 180^\circ \).
- \( m\angle 4 = 180^\circ - 114^\circ = 66^\circ \).

- Find \( m\angle 5 \):
- \( \angle 5 \) and \( \angle 9 \) are alternate interior angles.
- Therefore, \( m\angle 5 = m\angle 9 = 97^\circ \).

- Find \( m\angle 6 \):
- \( \angle 6 \) and \( \angle 9 \) are vertical angles.
- Therefore, \( m\angle 6 = m\angle 9 = 97^\circ \).

- Find \( m\angle 7 \):
- \( \angle 7 \) and \( \angle 12 \) are alternate interior angles.
- Therefore, \( m\angle 7 = m\angle 12 = 114^\circ \).

- Find \( m\angle 8 \):
- \( \angle 8 \) and \( \angle 12 \) are vertical angles.
- Therefore, \( m\angle 8 = m\angle 12 = 114^\circ \).

- Find \( m\angle 13 \):
- \( \angle 13 \) and \( \angle 9 \) are corresponding angles.
- Therefore, \( m\angle 13 = m\angle 9 = 97^\circ \).

- Find \( m\angle 14 \):
- \( \angle 14 \) and \( \angle 12 \) are corresponding angles.
- Therefore, \( m\angle 14 = m\angle 12 = 114^\circ \).

##### Final Answers for Problem 2:
\[
\begin{aligned}
&\text{a. } m\angle 1 = 97^\circ \\
&\text{b. } m\angle 2 = 83^\circ \\
&\text{c. } m\angle 3 = 114^\circ \\
&\text{d. } m\angle 4 = 66^\circ \\
&\text{e. } m\angle 5 = 97^\circ \\
&\text{f. } m\angle 6 = 97^\circ \\
&\text{g. } m\angle 7 = 114^\circ \\
&\text{h. } m\angle 8 = 114^\circ \\
&\text{k. } m\angle 13 = 97^\circ \\
&\text{l. } m\angle 14 = 114^\circ \\
\end{aligned}
\]

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Final Answer:

\[
\boxed{
\begin{aligned}
&\text{Problem 1:} \\
&\text{a. } 23^\circ, \text{b. } 23^\circ, \text{c. } 157^\circ, \text{d. } 23^\circ, \text{e. } 23^\circ, \text{f. } 23^\circ, \text{g. } 157^\circ \\
&\text{Problem 2:} \\
&\text{a. } 97^\circ, \text{b. } 83^\circ, \text{c. } 114^\circ, \text{d. } 66^\circ, \text{e. } 97^\circ, \text{f. } 97^\circ, \text{g. } 114^\circ, \text{h. } 114^\circ, \text{k. } 97^\circ, \text{l. } 114^\circ \\
\end{aligned}
}
\]