Find the value of x in each diagram where two parallel lines are cut by a transversal, using alternate angle properties.

To solve for the value of \( x \) in each figure, we will use the properties of alternate angles formed when two parallel lines are cut by a transversal. Here are the key properties:

1. Alternate Interior Angles: These are equal.
2. Alternate Exterior Angles: These are equal.
3. Corresponding Angles: These are equal.
4. Consecutive Interior Angles (Same-Side Interior Angles): These are supplementary (sum to 180°).

Let's solve each problem step by step.

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Problem 1


- The given angles are \( 146^\circ \) and \( 68^\circ \).
- The angle \( x \) is an alternate interior angle to \( 68^\circ \).
- Therefore, \( x = 68^\circ \).

Answer: \( x = 68^\circ \)

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Problem 2


- The given angles are \( 131^\circ \) and \( 49^\circ \).
- The angle \( x \) is an alternate exterior angle to \( 49^\circ \).
- Therefore, \( x = 49^\circ \).

Answer: \( x = 49^\circ \)

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Problem 3


- The given angles are \( 123^\circ \) and \( 53^\circ \).
- The angle \( x \) is an alternate interior angle to \( 53^\circ \).
- Therefore, \( x = 53^\circ \).

Answer: \( x = 53^\circ \)

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Problem 4


- The given angles are \( 62^\circ \) and \( 118^\circ \).
- The angle \( x \) is an alternate interior angle to \( 62^\circ \).
- Therefore, \( x = 62^\circ \).

Answer: \( x = 62^\circ \)

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Problem 5


- The given angles are \( 125^\circ \) and \( 55^\circ \).
- The angle \( x \) is an alternate interior angle to \( 55^\circ \).
- Therefore, \( x = 55^\circ \).

Answer: \( x = 55^\circ \)

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Problem 6


- The given angles are \( 102^\circ \) and \( 78^\circ \).
- The angle \( x \) is an alternate exterior angle to \( 78^\circ \).
- Therefore, \( x = 78^\circ \).

Answer: \( x = 78^\circ \)

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Problem 7


- The given angles are \( 151^\circ \) and \( 29^\circ \).
- The angle \( x \) is an alternate interior angle to \( 29^\circ \).
- Therefore, \( x = 29^\circ \).

Answer: \( x = 29^\circ \)

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Problem 8


- The given angles are \( 83^\circ \) and \( 97^\circ \).
- The angle \( x \) is an alternate interior angle to \( 97^\circ \).
- Therefore, \( x = 97^\circ \).

Answer: \( x = 97^\circ \)

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Problem 9


- The given angles are \( 130^\circ \) and \( 50^\circ \).
- The angle \( x \) is an alternate interior angle to \( 50^\circ \).
- Therefore, \( x = 50^\circ \).

Answer: \( x = 50^\circ \)

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Final Answers

\[
\boxed{
\begin{array}{ccc}
1) & x = 68^\circ & \\
2) & x = 49^\circ & \\
3) & x = 53^\circ & \\
4) & x = 62^\circ & \\
5) & x = 55^\circ & \\
6) & x = 78^\circ & \\
7) & x = 29^\circ & \\
8) & x = 97^\circ & \\
9) & x = 50^\circ & \\
\end{array}
}
\]