Parallel lines and transversals worksheet with algebraic expressions for angles.

1. For the first problem:
- Lines A and B are parallel, and C is a transversal.
- Angles 1 and 5 are corresponding angles, so they are equal.
- Angle 1 = (4x + 25)° and Angle 5 = (8x - 15)°.
- Set them equal: 4x + 25 = 8x - 15.
- Solve for x: 25 + 15 = 8x - 4x → 40 = 4x → x = 10.
- Angle 7 is vertical to Angle 5, so they are equal.
- Angle 5 = 8(10) - 15 = 80 - 15 = 65°.
- Therefore, Angle 7 = 65°.
- Final answers: x = 10, Angle 7 = 65°.

2. For the second problem:
- Angles 1 and 5 are corresponding angles, so they are equal.
- Angle 1 = (8x + 100)° and Angle 5 = (5x + 12)°.
- Set them equal: 8x + 100 = 5x + 12.
- Solve for x: 8x - 5x = 12 - 100 → 3x = -88 → x = -88/3.
- Angle 5 = 5(-88/3) + 12 = -440/3 + 12 = -440/3 + 36/3 = -404/3 ≈ -134.67°.
- This result is not possible for an angle measure, indicating a mistake.
- Recheck: The angles might be supplementary if they are same-side interior angles.
- Angles 1 and 5 are not same-side interior; they are corresponding.
- Alternatively, if the diagram shows angles 1 and 5 as same-side interior, they would be supplementary.
- But in standard labeling, 1 and 5 are corresponding.
- Recheck the equation: 8x + 100 = 5x + 12 → 3x = -88 → x = -88/3.
- This leads to a negative angle, which is not possible.
- The problem might have a typo, or the angles are not corresponding.
- Assume angles 1 and 5 are same-side interior angles (supplementary).
- Then: (8x + 100) + (5x + 12) = 180 → 13x + 112 = 180 → 13x = 68 → x = 68/13 ≈ 5.23.
- Angle 5 = 5(68/13) + 12 = 340/13 + 156/13 = 496/13 ≈ 38.15°.
- But this contradicts the corresponding angle assumption.
- Given the inconsistency, re-express the correct approach.
- Correct: Angles 1 and 5 are corresponding, so equal.
- 8x + 100 = 5x + 12 → 3x = -88 → x = -88/3.
- This is invalid.
- Likely, the angles are alternate interior or exterior.
- Angles 2 and 5 are alternate interior, so equal.
- Angle 2 = (8x + 100)°, Angle 5 = (5x + 12)°.
- 8x + 100 = 5x + 12 → 3x = -88 → x = -88/3.
- Still invalid.
- Angles 1 and 8 are alternate exterior, equal.
- Angle 1 = (8x + 100)°, Angle 8 = (5x + 12)°.
- 8x + 100 = 5x + 12 → 3x = -88 → x = -88/3.
- Same issue.
- The only possibility is that the angles are supplementary.
- Angles 1 and 6 are same-side interior, supplementary.
- Angle 1 = (8x + 100)°, Angle 6 = (5x + 12)°.
- 8x + 100 + 5x + 12 = 180 → 13x + 112 = 180 → 13x = 68 → x = 68/13.
- Angle 5 = 5(68/13) + 12 = 340/13 + 156/13 = 496/13 ≈ 38.15°.
- But the problem asks for Angle 5, which is (5x + 12)°.
- So Angle 5 = 496/13°.
- But this is not an integer.
- Given the confusion, assume the problem intends corresponding angles.
- The equation 8x + 100 = 5x + 12 has no valid solution.
- Therefore, there is likely an error in the problem statement.
- For the sake of proceeding, assume a different interpretation.
- Alternatively, angles 1 and 4 are supplementary (linear pair).
- But Angle 1 is given as (8x + 100)°, and Angle 4 is not given.
- This is not helpful.
- Re-express: The only way to get a valid solution is to assume that the angles are supplementary.
- But the problem states lines are parallel, so corresponding angles are equal.
- The given values lead to a contradiction.
- Therefore, the problem has an error.
- But for the purpose of the exercise, assume that the angles are supplementary.
- So x = 68/13, Angle 5 = 496/13°.
- However, this is not satisfactory.
- Given the complexity, focus on the other problems.

3. For the third problem:
- Angles 1 and 3 are vertical angles, so they are equal.
- Angle 1 = (2x + 12)°, Angle 3 = (6x + 104)°.
- Set them equal: 2x + 12 = 6x + 104.
- Solve for x: 12 - 104 = 6x - 2x → -92 = 4x → x = -23.
- Angle 1 = 2(-23) + 12 = -46 + 12 = -34°.
- Negative angle is not possible.
- This indicates a mistake.
- Angles 1 and 3 are not vertical; they are adjacent.
- In standard labeling, Angle 1 and Angle 3 are not vertical.
- Angles 1 and 3 are on the same side of the transversal, not opposite.
- Angles 1 and 4 are vertical.
- Angles 2 and 3 are vertical.
- So Angle 1 and Angle 3 are not vertical.
- Angles 1 and 4 are vertical.
- Angle 1 = (2x + 12)°, Angle 4 = (6x + 104)°.
- Set them equal: 2x + 12 = 6x + 104 → -92 = 4x → x = -23.
- Still negative.
- Angles 1 and 5 are corresponding, so equal.
- Angle 1 = (2x + 12)°, Angle 5 = (6x + 104)°.
- 2x + 12 = 6x + 104 → -92 = 4x → x = -23.
- Same issue.
- Angles 1 and 8 are alternate exterior, equal.
- Angle 1 = (2x + 12)°, Angle 8 = (6x + 104)°.
- 2x + 12 = 6x + 104 → -92 = 4x → x = -23.
- All lead to negative x.
- The only possibility is that the angles are supplementary.
- Angles 1 and 6 are same-side interior, supplementary.
- Angle 1 = (2x + 12)°, Angle 6 = (6x + 104)°.
- 2x + 12 + 6x + 104 = 180 → 8x + 116 = 180 → 8x = 64 → x = 8.
- Angle 1 = 2(8) + 12 = 16 + 12 = 28°.
- So x = 8, Angle 1 = 28°.
- This is valid.
- Therefore, the problem likely intends same-side interior angles to be supplementary.
- Final answers: x = 8, Angle 1 = 28°.

4. For the fourth problem:
- Angles 1 and 3 are vertical angles, so they are equal.
- Angle 1 = (9x - 21)°, Angle 3 = (7x - 5)°.
- Set them equal: 9x - 21 = 7x - 5.
- Solve for x: 9x - 7x = -5 + 21 → 2x = 16 → x = 8.
- Angle 3 = 7(8) - 5 = 56 - 5 = 51°.
- Final answers: x = 8, Angle 3 = 51°.

Final Answers:
1. x = 10, Angle 7 = 65°
2. x = 68/13, Angle 5 = 496/13° (but this is invalid; likely an error in the problem)
3. x = 8, Angle 1 = 28°
4. x = 8, Angle 3 = 51°