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

Let’s solve each problem step by step. Remember: Lines A and B are parallel, and line C is a transversal (a line that crosses both). We’ll use angle rules like:

- Vertical angles are equal (opposite angles when two lines cross).
- Corresponding angles are equal (same position on each parallel line).
- Alternate interior angles are equal (inside the parallels, opposite sides of transversal).
- Supplementary angles add up to 180° (when they form a straight line).

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Problem 1:

Given: One angle is 85°, next to Angle 1 on line A.

→ Angle 1 and 85° are on a straight line → they add to 180°.
So, Angle 1 = 180° - 85° = 95°

→ Angle 2 is directly below Angle 1, between the same lines — it’s an alternate interior angle with Angle 1? Wait — actually, look at positions.

Actually, Angle 2 is in the same relative position as the 85° angle but on the lower line — so it’s a corresponding angle to the 85° angle.

Wait — let me double-check.

In diagram 1:
- The 85° angle is above line A, right side of transversal.
- Angle 1 is adjacent to it on the left — so yes, supplementary → 95°.
- Angle 2 is below line B, left side of transversal — that’s vertically opposite to the angle that corresponds to Angle 1.

Better way: Since lines are parallel, corresponding angles are equal.

The angle that matches Angle 2 is the one directly across from the 85° angle — which is also 85° (vertical angle), and since it’s corresponding to Angle 2, then Angle 2 = 85°.

Alternatively: Angle 1 and Angle 2 are alternate interior angles? Let’s see:

Angle 1 is top-left inside the “Z” shape? Actually, no — better to label:

Top intersection: angles around point — 85° given, so its vertical angle is also 85°, and the other two are 95° each.

Bottom intersection: Angle 2 is in the same position as the 95° angle on top? No.

Actually, simplest:

Angle 1 and the 85° angle are adjacent on a straight line → sum to 180° → Angle 1 = 95°.

Angle 2 is corresponding to the 85° angle (both are on the right side of transversal, one above top line, one below bottom line) → so Angle 2 = 85°.

Yes.

✔ Angle 1 = 95°
✔ Angle 2 = 85°

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Problem 2:

Given: 122° angle on line A, right side.

Angle 1 is adjacent to it on the left → straight line → 180° - 122° = 58°

Angle 2 is on line B, left side — same side as Angle 1, but below.

Since lines are parallel, Angle 2 and Angle 1 are alternate interior angles? Let’s check positions.

Actually, Angle 2 is in the same position as the 122° angle? No.

Wait — the 122° angle is above line A, right side.

Its vertical angle is also 122°, below line A, left side — that would be corresponding to Angle 2? Not quite.

Better: Angle 2 and the 122° angle are on the same side of the transversal, but one above top line, one below bottom line — not corresponding.

Actually, Angle 2 is vertically opposite to the angle that is corresponding to the 58° angle.

Easier: Angle 1 = 58° (as calculated).

Now, Angle 2 is on the lower line, left side — same side as Angle 1, but below. So they are consecutive interior angles? Or alternate?

Actually, if you look, Angle 1 and Angle 2 are on opposite sides of the transversal and both inside the parallel lines — so they are alternate interior angles → should be equal.

Is that right? In standard terms:

- Alternate interior angles: inside the parallels, on opposite sides of transversal.

Here, Angle 1 is top-left inside, Angle 2 is bottom-right inside? Wait, no — in diagram 2:

Transversal goes from bottom-left to top-right.

Line A (top): angle labeled 122° is on the right side, above line A.

Then Angle 1 is on the left side, above line A — so adjacent to 122° → 58°.

Line B (bottom): Angle 2 is on the left side, below line B.

So Angle 1 is above top line, left side; Angle 2 is below bottom line, left side — so they are corresponding angles! Both on left side, one above top, one below bottom.

Therefore, Angle 2 = Angle 1 = 58°.

But wait — is that correct? Corresponding angles are in matching corners.

Standard definition: corresponding angles are in the same relative position at each intersection.

At top intersection: Angle 1 is upper-left.

At bottom intersection: Angle 2 is lower-left — that’s not the same relative position.

Upper-left at top vs lower-left at bottom — those are not corresponding.

Actually, the corresponding angle to Angle 1 (upper-left at top) would be upper-left at bottom — but that’s not labeled.

Perhaps I need to think differently.

Let me define:

At top intersection (line A and C):

- Given angle: 122° — let's say it's the angle between line A and C, on the right side, above line A.

Then the angle adjacent to it on the left (Angle 1) is 180° - 122° = 58°.

This 58° angle is between line A and C, on the left side, above line A.

Now, at bottom intersection (line B and C):

Angle 2 is between line B and C, on the left side, below line B.

These two angles (58° above top line left, and Angle 2 below bottom line left) are actually vertically opposite to angles that are corresponding.

Note that the angle directly below the 58° angle (at top intersection) is also 58° (vertical angle), and that angle is between line A and C, left side, below line A — which is an interior angle.

Similarly, at bottom, the angle above line B, left side, is the corresponding angle to that — so it should also be 58°.

But Angle 2 is below line B, left side — which is vertically opposite to that 58° angle — so Angle 2 = 58°.

Yes! So Angle 2 = 58°.

Alternatively, since the 122° angle has a corresponding angle on the bottom line — which would be 122° on the right side, below line B — and Angle 2 is adjacent to that on the left, so 180° - 122° = 58°.

Same result.

✔ Angle 1 = 58°
✔ Angle 2 = 58°

Wait — is that possible? Both 58°? Let me confirm with another method.

The angle vertically opposite to 122° is also 122° (below line A, left side). This is an interior angle on the top.

The consecutive interior angle on the bottom would be on the same side — so below line B, left side — which is Angle 2. Consecutive interior angles add to 180°.

So 122° + Angle 2 = 180° → Angle 2 = 58°.

And Angle 1 is 58° as before.

Yes.

So both are 58°.

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Problem 3:

Given: 72° angle on line A, right side, above.

Angle 1 is adjacent to it on the left → straight line → 180° - 72° = 108°

Angle 2 is on line B, left side, below.

Similar to problem 2.

Angle 1 = 108° (above top line, left side).

The corresponding angle on the bottom would be above bottom line, left side — which should also be 108°.

Then Angle 2 is vertically opposite to that — so also 108°.

Or: the 72° angle has a corresponding angle on the bottom: below bottom line, right side — 72°.

Then Angle 2 is adjacent to that on the left → 180° - 72° = 108°.

Yes.

✔ Angle 1 = 108°
✔ Angle 2 = 108°

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Problem 4:

Given: 140° angle on line B, right side, below.

Angle 2 is adjacent to it on the left → straight line → 180° - 140° = 40°

Angle 1 is on line A, right side, above.

Now, the 140° angle is below line B, right side.

Its vertical angle is above line B, left side — 140°.

But we need relation to Angle 1.

Note that Angle 2 = 40° (above line B, right side? Wait.

Diagram 4: transversal goes from top-left to bottom-right.

Line B (bottom): 140° is given on the right side, below line B.

So adjacent angle on the left, below line B, is 40° — but that’s not Angle 2.

Looking at diagram: Angle 2 is labeled on line B, left side, above line B? Or below?

In the image description: for problem 4, it says "140°" near line B, and Angle 2 is between line B and C, on the left side.

Assuming standard labeling: the 140° is the angle between line B and C, on the right side, below line B.

Then the angle adjacent to it on the left, below line B, is 40°.

But Angle 2 is probably the angle between line B and C, on the left side, above line B — because typically angles are measured inside or as shown.

To avoid confusion, let's think:

The 140° angle and Angle 2 are on a straight line? If they are adjacent on line B, then yes.

In most such diagrams, when an angle is given like that, and another angle is labeled nearby on the same line, they are adjacent.

So likely, Angle 2 and 140° are adjacent on line B → sum to 180° → Angle 2 = 40°.

Now, Angle 1 is on line A, right side, above.

What is the relationship?

The angle corresponding to the 140° angle: 140° is below line B, right side.

Corresponding angle would be below line A, right side — but that’s not labeled.

The vertical angle to 140° is above line B, left side — 140°.

Then, the alternate interior angle to that would be above line A, right side — which is Angle 1.

Alternate interior angles: inside the parallels, opposite sides of transversal.

Above line B, left side (140°) and above line A, right side (Angle 1) — are they alternate interior?

Actually, "interior" usually means between the two parallel lines.

Above line A is exterior, below line B is exterior.

Between line A and B is interior.

So perhaps not.

Better: the angle that is vertically opposite to Angle 1 is below line A, left side.

That angle and the 140° angle (below line B, right side) are not directly related.

Note that the 140° angle and the angle above line B, right side, are adjacent → so above line B, right side is 40°.

This 40° angle is between line B and C, above line B, right side.

Now, this is an interior angle.

The corresponding angle on top would be above line A, right side — which is Angle 1.

Corresponding angles: same relative position.

Above line B, right side and above line A, right side — yes, corresponding.

So Angle 1 = 40°.

But earlier I said Angle 2 = 40°.

Let me confirm Angle 2.

If 140° is below line B, right side, then the angle adjacent on the left, below line B, is 40° — but is that Angle 2?

In the diagram, Angle 2 is likely the angle between line B and C, on the left side, and since the transversal is going down to the right, "left side" might mean the acute angle.

Typically in such problems, the labeled angles are the ones formed, and Angle 2 is probably the one that makes sense with the transversal.

From the calculation: if Angle 2 is adjacent to 140° on the straight line, then Angle 2 = 40°.

And Angle 1 is corresponding to the 40° angle that is above line B, right side — which is vertically opposite to the angle adjacent to 140° on the top side.

Specifically:

- At bottom intersection: 140° below right → so above right is 40° (since straight line).

- This 40° above right on bottom line corresponds to Angle 1 above right on top line → so Angle 1 = 40°.

Yes.

So both Angle 1 and Angle 2 are 40°? That seems odd, but mathematically possible.

Angle 2: if it's the angle on the left side of transversal at bottom, and if 140° is on the right, then depending on which side Angle 2 is.

In standard interpretation, for problem 4:

The 140° is given, and Angle 2 is next to it on the same line, so likely supplementary → 40°.

Angle 1 is on the top, and since the lines are parallel, and the transversal, the angle that is alternate exterior or something.

Another way: the 140° angle and Angle 1 are on the same side of the transversal, but one below bottom, one above top — so they are not directly related, but their supplements might be.

Note that the angle vertically opposite to 140° is 140° (above line B, left side).

Then, this 140° and Angle 1 are consecutive exterior angles or something.

Actually, Angle 1 and the 140° angle (above line B, left side) are on the same side of the transversal, and both outside the parallels — so they are consecutive exterior angles, which add to 180° if lines are parallel.

Is that a rule? Yes, consecutive exterior angles are supplementary when lines are parallel.

So Angle 1 + 140° = 180° → Angle 1 = 40°.

And Angle 2: if it's the angle adjacent to 140° on the straight line, then Angle 2 = 40°.

But in the diagram, Angle 2 might be the other angle.

Let's assume based on common labeling.

In many textbooks, for such a diagram, if 140° is given at bottom right below, then Angle 2 is often the angle at bottom left above or something.

To resolve, let's think logically.

Suppose at bottom intersection:

- Angle between line B and C, below line B, right side: 140°

- Then, angle between line B and C, below line B, left side: 40° (since 180° - 140°)

- Angle between line B and C, above line B, right side: 40° (vertical to the 40° below left? No.

Vertical angles: the angle opposite to 140° is above line B, left side — 140°.

The angle opposite to the 40° below left is above line B, right side — 40°.

So at bottom:

- Above line B, right side: 40°

- Above line B, left side: 140°

- Below line B, right side: 140°

- Below line B, left side: 40°

Now, Angle 2 is labeled — in the problem, it's probably the angle that is between line B and C, and since it's called "Angle 2", and in context, likely it's the one that is not the given 140°, and adjacent or something.

In the user's image description, for problem 4, it says "140°" and then "Angle 2" is separate, so likely Angle 2 is the adjacent angle on the straight line, so 40°.

Similarly, Angle 1 is on top.

Now, the angle above line A, right side (Angle 1) corresponds to the angle above line B, right side, which is 40°.

So Angle 1 = 40°.

Thus, both are 40°.

But let's verify with another pair.

The angle above line A, left side would correspond to above line B, left side, which is 140°, so that would be 140°, but that's not labeled.

So for the labeled angles, Angle 1 = 40°, Angle 2 = 40°.

Perhaps Angle 2 is meant to be the other angle.

In some diagrams, Angle 2 might be the interior angle.

But based on the pattern of previous problems, and to be consistent, I'll go with:

Angle 2 = 180° - 140° = 40° (adjacent on straight line)

Angle 1 = corresponding to the 40° angle at bottom (above line B, right side) = 40°

So yes.

✔ Angle 1 = 40°
✔ Angle 2 = 40°

But let me double-check problem 4 with a different approach.

The 140° angle and Angle 1 are on the same side of the transversal. The 140° is exterior below, Angle 1 is exterior above. For parallel lines, consecutive exterior angles are supplementary.

Consecutive exterior angles: both outside, same side of transversal.

Here, 140° is below line B, right side — exterior.

Angle 1 is above line A, right side — also exterior, same side (right side of transversal).

So yes, they are consecutive exterior angles → sum to 180°.

So Angle 1 + 140° = 180° → Angle 1 = 40°.

For Angle 2: if it's the angle at bottom, left side, and if it's the one adjacent to 140° on the line, then it's 40°, but is it on the same side?

In the diagram, Angle 2 is likely the angle between line B and C, on the left side, and since the transversal is slanting, it could be the acute angle.

Moreover, in the answer format, it's expected to be a number, and 40° makes sense.

Perhaps Angle 2 is vertically opposite or something, but I think 40° is correct.

So final answers:

Problem 1: Angle 1 = 95°, Angle 2 = 85°
Problem 2: Angle 1 = 58°, Angle 2 = 58°
Problem 3: Angle 1 = 108°, Angle 2 = 108°
Problem 4: Angle 1 = 40°, Angle 2 = 40°

But in problem 2 and 3, both angles are equal, which is fine.

Now, to box the final answer as per instruction.

Final Answer:
1. Angle 1 = 95°, Angle 2 = 85°
2. Angle 1 = 58°, Angle 2 = 58°
3. Angle 1 = 108°, Angle 2 = 108°
4. Angle 1 = 40°, Angle 2 = 40°