Parallel and Perpendicular Lines with a Trasnversal Line worksheet ... - Free Printable

Let's solve each problem step by step using the properties of parallel lines and a transversal.

We are given that:
- Lines A and B are parallel.
- Line C is a transversal, cutting through both parallel lines.

When a transversal intersects two parallel lines, several angle relationships apply:
1. Corresponding angles are equal.
2. Alternate interior angles are equal.
3. Alternate exterior angles are equal.
4. Consecutive interior angles (same-side interior) are supplementary (add to 180°).
5. Vertical angles are equal.

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Problem 1

Given:
Angle at the top = 85°

We need to find:
- Angle 1
- Angle 2

From the diagram:
- Angle 1 is vertically opposite to the 85° angle → So Angle 1 = 85°
- Angle 2 is alternate interior to the 85° angle → Since A || B, alternate interior angles are equal → Angle 2 = 85°

✔ Answer:
- Angle 1 = 85°
- Angle 2 = 85°

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Problem 2

Given:
Angle between line B and transversal = 122°

We need to find:
- Angle 1
- Angle 2

From the diagram:
- The 122° angle is on line B, and it’s corresponding to Angle 2 (on line A), so:
- Angle 2 = 122° (corresponding angles)
- Now, Angle 1 is adjacent to Angle 2 on a straight line → they form a linear pair → sum to 180°
- So, Angle 1 = 180° – 122° = 58°

Alternatively, Angle 1 is alternate interior to the angle adjacent to 122° on line B. The adjacent angle is 180° – 122° = 58°, and since alternate interior angles are equal, Angle 1 = 58°.

✔ Answer:
- Angle 1 = 58°
- Angle 2 = 122°

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Problem 3

Given:
Angle on line A = 72°

We need to find:
- Angle 1
- Angle 2

From the diagram:
- Angle 1 is vertically opposite to the 72° angle → So Angle 1 = 72°
- Angle 2 is alternate interior to the 72° angle → So Angle 2 = 72°

✔ Answer:
- Angle 1 = 72°
- Angle 2 = 72°

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Problem 4

Given:
Angle on line B = 140°

We need to find:
- Angle 1
- Angle 2

From the diagram:
- The 140° angle is on line B, and it’s corresponding to Angle 2 → So Angle 2 = 140°
- Angle 1 is adjacent to Angle 2 on a straight line → forms a linear pair → sum to 180°
- So, Angle 1 = 180° – 140° = 40°

Alternatively, Angle 1 is alternate interior to the angle adjacent to 140° on line B. That adjacent angle is 40°, so Angle 1 = 40°.

✔ Answer:
- Angle 1 = 40°
- Angle 2 = 140°

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✔ Final Answers:

| Problem | Angle 1 | Angle 2 |
|--------|---------|---------|
| 1 | 85° | 85° |
| 2 | 58° | 122° |
| 3 | 72° | 72° |
| 4 | 40° | 140° |

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🔍 Summary of Concepts Used:

- Vertical angles: Equal
- Corresponding angles: Equal (when lines are parallel)
- Alternate interior angles: Equal (when lines are parallel)
- Linear pairs: Add to 180°

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