Parallel lines and transversals interactive worksheet | Live ... - Free Printable

Problem Analysis and Solution

The problems involve finding angles, solving for variables using properties of parallel lines, transversals, and basic angle relationships. Let's solve each problem step by step.

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#### Part 1: Find \( m\angle1 \) and \( m\angle2 \)

##### Problem 1
- Given: \( \angle1 = 60^\circ \)
- Since \( \angle1 \) and \( \angle2 \) are vertical angles, they are equal.
- Therefore, \( m\angle2 = 60^\circ \).

Answer:
\[ m\angle1 = 60^\circ, \quad m\angle2 = 60^\circ \]

##### Problem 2
- Given: \( \angle2 = 40^\circ \)
- Since \( \angle1 \) and \( \angle2 \) are vertical angles, they are equal.
- Therefore, \( m\angle1 = 40^\circ \).

Answer:
\[ m\angle1 = 40^\circ, \quad m\angle2 = 40^\circ \]

##### Problem 3
- Given: One angle is \( 45^\circ \).
- Since the two lines are parallel and the given angle is an alternate interior angle to \( \angle1 \), \( \angle1 = 45^\circ \).
- \( \angle2 \) is a corresponding angle to the given \( 45^\circ \) angle, so \( \angle2 = 45^\circ \).

Answer:
\[ m\angle1 = 45^\circ, \quad m\angle2 = 45^\circ \]

##### Problem 4
- Given: One angle is \( 75^\circ \).
- Since the two lines are parallel, \( \angle2 \) is a corresponding angle to the given \( 75^\circ \) angle, so \( \angle2 = 75^\circ \).
- \( \angle1 \) is an alternate interior angle to the given \( 75^\circ \) angle, so \( \angle1 = 75^\circ \).

Answer:
\[ m\angle1 = 75^\circ, \quad m\angle2 = 75^\circ \]

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#### Part 2: Find the values of \( x \) and \( y \)

##### Problem 5
- The diagram shows intersecting lines with angles labeled \( x^\circ \), \( y^\circ \), and \( 80^\circ \).
- Vertical angles are equal:
- \( x = 80^\circ \)
- Adjacent angles form a linear pair (sum to \( 180^\circ \)):
- \( y + 80^\circ = 180^\circ \)
- \( y = 100^\circ \)

Answer:
\[ x = 80, \quad y = 100 \]

##### Problem 6
- The diagram shows a perpendicular line intersecting two parallel lines.
- The angle marked \( x^\circ \) is a right angle because the lines are perpendicular:
- \( x = 90^\circ \)
- The angle marked \( y^\circ \) is also a right angle (corresponding or alternate interior angle):
- \( y = 90^\circ \)

Answer:
\[ x = 90, \quad y = 90 \]

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#### Part 3: Find the value of \( x \)

##### Problem 7
- The diagram shows a transversal intersecting two parallel lines.
- The angle marked \( 2x^\circ \) is supplementary to the given \( 80^\circ \) angle:
- \( 2x + 80^\circ = 180^\circ \)
- \( 2x = 100^\circ \)
- \( x = 50^\circ \)

Answer:
\[ x = 50 \]

##### Problem 8
- The diagram shows a transversal intersecting two parallel lines.
- The angle marked \( 2x^\circ \) is supplementary to the given \( 110^\circ \) angle:
- \( 2x + 110^\circ = 180^\circ \)
- \( 2x = 70^\circ \)
- \( x = 35^\circ \)

Answer:
\[ x = 35 \]

##### Problem 9
- The diagram shows a transversal intersecting two parallel lines.
- The angle marked \( (x - 10)^\circ \) is supplementary to the given \( 100^\circ \) angle:
- \( (x - 10) + 100^\circ = 180^\circ \)
- \( x - 10 = 80^\circ \)
- \( x = 90^\circ \)

Answer:
\[ x = 90 \]

##### Problem 10
- The diagram shows a transversal intersecting two parallel lines.
- The angle marked \( (3x + 15)^\circ \) is supplementary to the given \( 60^\circ \) angle:
- \( (3x + 15) + 60^\circ = 180^\circ \)
- \( 3x + 75^\circ = 180^\circ \)
- \( 3x = 105^\circ \)
- \( x = 35^\circ \)

Answer:
\[ x = 35 \]

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Final Answers

1. \( m\angle1 = 60^\circ, \quad m\angle2 = 60^\circ \)
2. \( m\angle1 = 40^\circ, \quad m\angle2 = 40^\circ \)
3. \( m\angle1 = 45^\circ, \quad m\angle2 = 45^\circ \)
4. \( m\angle1 = 75^\circ, \quad m\angle2 = 75^\circ \)
5. \( x = 80, \quad y = 100 \)
6. \( x = 90, \quad y = 90 \)
7. \( x = 50 \)
8. \( x = 35 \)
9. \( x = 90 \)
10. \( x = 35 \)

\[
\boxed{
\begin{array}{l}
1. m\angle1 = 60^\circ, \quad m\angle2 = 60^\circ \\
2. m\angle1 = 40^\circ, \quad m\angle2 = 40^\circ \\
3. m\angle1 = 45^\circ, \quad m\angle2 = 45^\circ \\
4. m\angle1 = 75^\circ, \quad m\angle2 = 75^\circ \\
5. x = 80, \quad y = 100 \\
6. x = 90, \quad y = 90 \\
7. x = 50 \\
8. x = 35 \\
9. x = 90 \\
10. x = 35 \\
\end{array}
}
\]