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

Problem Analysis and Solution

The problems involve finding angles and solving for variables using properties of parallel lines, transversals, and 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 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 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 two 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 corresponding angle to the \( 90^\circ \) angle formed by the perpendicular line, so \( x = 90^\circ \).
- The angle marked \( y^\circ \) is also a corresponding angle to the \( 90^\circ \) angle, so \( y = 90^\circ \).

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 an alternate interior angle to the \( 80^\circ \) angle, so:
\[ 2x = 80 \]
\[ x = 40 \]

Answer:
\[ x = 40 \]

##### Problem 8
- The diagram shows a transversal intersecting two parallel lines.
- The angle marked \( 2x^\circ \) and the \( 110^\circ \) angle are supplementary (they form a linear pair):
\[ 2x + 110 = 180 \]
\[ 2x = 70 \]
\[ x = 35 \]

Answer:
\[ x = 35 \]

##### Problem 9
- The diagram shows a transversal intersecting two parallel lines.
- The angle marked \( (x - 10)^\circ \) and the \( 100^\circ \) angle are corresponding angles, so:
\[ x - 10 = 100 \]
\[ x = 110 \]

Answer:
\[ x = 110 \]

##### Problem 10
- The diagram shows a transversal intersecting two parallel lines.
- The angle marked \( (3x + 15)^\circ \) and the \( 60^\circ \) angle are corresponding angles, so:
\[ 3x + 15 = 60 \]
\[ 3x = 45 \]
\[ x = 15 \]

Answer:
\[ x = 15 \]

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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. \( x = 40 \)
8. \( x = 35 \)
9. \( x = 110 \)
10. \( x = 15 \)

\[
\boxed{
\begin{array}{ll}
1. & m\angle1 = 60^\circ, \quad m\angle2 = 60^\circ \\
2. & m\angle1 = 40^\circ, \quad m\angle2 = 40^\circ \\
3. & m\angle1 = 45^\circ, \quad m\angle2 = 45^\circ \\
4. & m\angle1 = 75^\circ, \quad m\angle2 = 75^\circ \\
5. & x = 80, \quad y = 100 \\
6. & x = 90, \quad y = 90 \\
7. & x = 40 \\
8. & x = 35 \\
9. & x = 110 \\
10. & x = 15 \\
\end{array}
}
\]