Geometry worksheet on angles formed by parallel lines and a transversal.

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 C = 122°

We need to find:
- Angle 1
- Angle 2

Looking at the diagram:
- The 122° angle is on line B.
- Angle 2 is adjacent to this 122° angle, forming a straight line → so they are supplementary.
- Therefore, Angle 2 = 180° − 122° = 58°
- Angle 1 is corresponding to Angle 2 (since A || B), or it’s alternate interior to the 122° angle? Let's check:

Actually:
- The 122° angle and Angle 1 are on opposite sides of the transversal and between the parallel lines → they are alternate interior angles → so Angle 1 = 122°

Wait — no! Let’s be careful.

Actually, the 122° angle is on line B, and Angle 1 is on line A, same side of the transversal, but above.

But look: the 122° angle and Angle 1 are corresponding angles if we consider the direction.

Wait — let's clarify:

The 122° angle is below the transversal on line B.
Angle 1 is above the transversal on line A.

So they are not corresponding.

Instead, note:
- The 122° angle and Angle 2 are adjacent angles on a straight line → so:
- Angle 2 = 180° − 122° = 58°

Now, Angle 1 and Angle 2 are corresponding angles (both are on the same side of the transversal, and above their respective lines) → since A || B, corresponding angles are equal → Angle 1 = 58°

Alternatively, you can think:
- Angle 1 and the 122° angle are consecutive interior angles → they add to 180° → so Angle 1 = 180° − 122° = 58°

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

Wait — actually, Angle 2 is adjacent to 122° → so yes, Angle 2 = 58°
And Angle 1 is corresponding to Angle 2 → so also 58°

Yes.

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

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Problem 3

Given:
Angle on line A = 72°

We need to find:
- Angle 1
- Angle 2

From the diagram:
- The 72° angle is on line A, above the transversal.
- Angle 1 is below the transversal on line B → it is corresponding to the 72° angle → so Angle 1 = 72°
- Angle 2 is adjacent to Angle 1 → forms a straight line → so Angle 2 = 180° − 72° = 108°

Alternatively:
- Angle 2 is alternate interior to the 72° angle? No — not directly.

Wait:
- The 72° angle and Angle 2 are on opposite sides of the transversal, but one is above, one is below, and between the lines → so they are alternate interior angles → so Angle 2 = 72°?

No — wait: the 72° angle is on line A, above the transversal.

Angle 2 is on line B, below the transversal → so they are not alternate interior.

Alternate interior angles are on opposite sides of the transversal and between the lines.

So:
- The 72° angle is on line A, above transversal.
- The angle below the transversal on line A is vertical to the 72° angle → 180° − 72° = 108°?
No — vertical angles are equal.

Wait — the 72° angle has a vertical angle directly opposite it on line A → which is also 72°.

Then, that 72° angle (on line A, below transversal) and Angle 2 (on line B, below transversal) are corresponding angles → so Angle 2 = 72°?

But no — the 72° angle is on line A, above.

Let me re-label mentally:

- Transversal crosses line A: forms 72° angle (top-right)
- Then crosses line B: forms Angle 1 (bottom-left), Angle 2 (bottom-right)

Actually, from the diagram:
- The 72° angle is on the top right of line A.
- Angle 1 is on the bottom left of line B → this is alternate interior to the 72° angle → so Angle 1 = 72°
- Angle 2 is adjacent to Angle 1 → so Angle 2 = 180° − 72° = 108°

Alternatively, Angle 2 is corresponding to the angle below the transversal on line A → which is vertical to 72° → so that angle is 72°, and its corresponding angle on line B is Angle 2 → so Angle 2 = 72°? That would conflict.

Wait — let's define:

- The 72° angle is above line A, on the right side of transversal.
- Its vertical angle is below line A, on the left side → also 72°.
- This 72° angle (below A, left) and Angle 1 (below B, left) are corresponding angles → so Angle 1 = 72°
- Angle 2 is next to Angle 1 on the same line B → so they form a straight line → Angle 2 = 180° − 72° = 108°

✔ Answer:
- Angle 1 = 72°
- Angle 2 = 108°

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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 below line B, on the left side of transversal.
- Angle 2 is above line B, on the left side → so it's adjacent to the 140° angle → so:
- Angle 2 = 180° − 140° = 40°
- Now, Angle 1 is on line A, above the transversal, on the left side → this is corresponding to Angle 2 → so Angle 1 = 40°

Alternatively:
- Angle 1 and the 140° angle are consecutive interior angles → they should add to 180° → so Angle 1 = 180° − 140° = 40°

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

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✔ Final Answers:

#### Problem 1
- Angle 1 = 85°
- Angle 2 = 85°

#### Problem 2
- Angle 1 = 58°
- Angle 2 = 58°

#### Problem 3
- Angle 1 = 72°
- Angle 2 = 108°

#### Problem 4
- Angle 1 = 40°
- Angle 2 = 40°

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Let me know if you'd like a visual explanation or diagram labels!