Parallel Lines and Transversal worksheet - Free Printable

Let's solve each of these geometry problems step by step. All of them involve parallel lines cut by a transversal, and we'll use angle relationships such as:

- Corresponding angles are equal.
- Alternate interior angles are equal.
- Same-side interior angles are supplementary (add to 180°).
- Vertical angles are equal.
- Linear pairs add to 180°.

We'll go through each question one at a time.

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13)

We have two parallel lines cut by a transversal. One angle is given as 58°, and we're asked to find the angle marked with a question mark.

Looking at the diagram:
- The 58° angle is on the lower line, on the right side.
- The angle marked "?" is on the upper line, on the left side — it's vertically opposite to the corresponding angle above the 58°.

But let’s think carefully:
The 58° angle and the angle directly above it (on the same side of the transversal) form a linear pair with the angle across from it.

Wait — actually, since the lines are parallel, and the transversal cuts them:

The 58° angle and the angle marked "?" are corresponding angles? No — they are not in the same relative position.

Actually, the angle marked "?" is opposite to the 58° angle across the transversal — so they are vertical angles?

No — vertical angles are formed at the same intersection.

Wait — there are two intersections here: one on the top line, one on the bottom.

The 58° angle is on the bottom line, and the "?" is on the top line.

So the 58° angle and the angle directly above it (on the top line) are corresponding angles, so they should be equal.

But the "?" is not that one — it's the other angle on the top line.

Wait — let's clarify:

At the bottom intersection:
- One angle is 58°
- The angle adjacent to it (forming a linear pair) is 180° - 58° = 122°

Now, the angle marked "?" is on the top line, on the left side, and appears to be alternate interior or corresponding?

Actually, looking at the direction of the transversal:

The transversal goes from top-left to bottom-right.

So:
- The 58° angle is on the bottom line, on the right side, inside the two lines.
- The angle "?" is on the top line, on the left side, inside the two lines.

So these two are alternate interior angles → they are equal?

Wait — no: alternate interior angles are on opposite sides of the transversal and between the lines.

So yes, if the 58° is on the right, then the alternate interior angle would be on the left, between the lines, on the top line.

Yes! So "?" is an alternate interior angle to the 58° angle.

Therefore, ? = 58°

✔ Answer: C) 58°

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14)

Given: 60° angle on the top line, on the right side, outside the lines.

The angle marked "?" is on the bottom line, on the left side, inside the lines.

Let’s analyze:

- The 60° angle is on the top line, and it's on the right, outside the parallel lines.
- The angle "?" is on the bottom line, on the left, inside.

So, the 60° angle and the angle directly below it (on the bottom line, right side) are corresponding angles, so they are equal → 60°.

But "?" is on the left side of the bottom line.

So, at the bottom intersection:
- The angle on the right is 60° (corresponding)
- Then the angle on the left (adjacent) is supplementary: 180° - 60° = 120°

But wait — is "?" that angle?

No — look: the angle "?" is on the left, but is it inside or outside?

From the diagram: the red dot indicates the direction of the arrow, and the angle is drawn on the left side, between the lines.

So it's interior, on the left.

So the 60° is on the right, exterior, on the top.

Then, the angle on the bottom line, left, interior — how is this related?

Let’s find the angle adjacent to the 60° angle.

At the top line, the 60° angle is on the right, outside. Its supplementary angle (on the same line) is 180° - 60° = 120°, which is interior, on the right side.

This 120° angle is same-side interior to the angle "?".

Wait — no: the angle "?" is on the left.

So perhaps better: Let’s use vertical angles and corresponding.

Alternatively: the 60° angle has a vertical angle on the same side, which is also 60°, but on the other side of the transversal.

But more useful: the angle on the bottom line, on the right, interior, is corresponding to the top line, right, interior angle.

But the top line, right, interior angle is 180° - 60° = 120° (since 60° is exterior).

So corresponding angle on bottom line, right, interior = 120°

Then, the angle "?" is on the left, interior, so it's alternate interior to the 120° angle? No.

Wait: the interior angles on the same side of the transversal are supplementary.

So the interior angles on the right are both 120°? Wait — no.

Wait: the top line, interior, right = 180° - 60° = 120°

Then, the bottom line, interior, right = corresponding = 120°

Then, the bottom line, interior, left = ? It's adjacent to that 120°, so it's 180° - 120° = 60°

But wait — that can't be, because then it would be 60°, but let's see.

Actually, the interior angles on the same side of the transversal are supplementary.

So the top interior right = 120°, and the bottom interior left is not on the same side.

Wait — same-side interior angles: both on the same side of the transversal and between the lines.

So:
- Top line, interior, right = 120°
- Bottom line, interior, right = 120° → corresponding → yes
- Then the bottom line, interior, left is adjacent to that → 180° - 120° = 60°

But that 60° is on the left, interior, bottom — which is exactly where "?" is.

So "?" = 60°

Wait — but is that correct?

Wait: the 60° angle is exterior, on the top, right.

Its vertical angle is also 60°, on the bottom, left, exterior?

Wait — let’s do it properly.

Let’s label the angles.

At the top intersection:
- One angle is 60° (exterior, right)
- Adjacent to it (on the same line) is 180° - 60° = 120° (interior, right)
- Vertical angle to 60° is 60° (exterior, left)
- Vertical angle to 120° is 120° (interior, left)

Wait — no: vertical angles are opposite.

So at the top:
- 60° (right, exterior)
- Opposite to it (left, exterior) = 60°
- The angle between the lines on the right = 120°
- On the left, between the lines = 120°

Now at the bottom intersection:
- The corresponding angles:
- Right, exterior: corresponds to top right exterior = 60°
- Left, exterior: corresponds to top left exterior = 60°
- Right, interior: corresponds to top right interior = 120°
- Left, interior: corresponds to top left interior = 120°

But the angle "?" is on the bottom line, left, interior → so it should be 120°?

But that's not among the choices.

Choices: A) 46° B) 130° C) 60° D) 116°

Wait — none of them is 120°.

Wait — maybe I misread.

Wait — the angle "?" is not the interior left — look again.

In problem 14, the angle "?" is on the bottom line, on the left side, but it's not between the lines — it's outside?

Wait — the diagram shows the transversal going up and to the right.

On the bottom line, the angle "?" is on the left, and the arrow is pointing left, so the angle is formed between the transversal and the line, on the left side, and it's outside the two parallel lines.

So it's exterior, on the left.

And at the top, the 60° is exterior, on the right.

So the angle "?" is on the bottom, left, exterior.

Now, what is its relationship?

The top, right, exterior = 60°

The bottom, left, exterior is opposite to it? Not vertically.

But notice: the top, right, exterior and the bottom, left, exterior are alternate exterior angles?

Wait — alternate exterior angles are on opposite sides of the transversal and outside the lines.

Yes!

So:
- Top, right, exterior = 60°
- Bottom, left, exterior = alternate exterior → equal → 60°

So "?" = 60°

✔ Answer: C) 60°

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15)

Given: 54° angle on the bottom line, on the right, interior.

Angle "?" is on the top line, on the left, interior.

So, both are interior, on opposite sides of the transversal → alternate interior angles → equal.

So "?" = 54°

✔ Answer: D) 54°

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16)

Given: 135° angle on the bottom line, on the right, interior.

Angle "?" is on the top line, on the left, interior.

So, same as above: alternate interior angles → should be equal?

Wait — 135° and "?" are on opposite sides, between the lines → yes, alternate interior angles → equal.

So "?" = 135°?

But 135° is not among the options.

Options: A) 45° B) 102° C) 42° D) 53°

Hmm — something’s wrong.

Wait — let’s check the diagram.

The 135° is on the bottom line, on the right, interior.

Then the alternate interior angle would be on the top line, on the left, interior — which is exactly where "?" is.

So it should be 135° — but that's not an option.

Wait — maybe it's not alternate interior?

Wait — could it be same-side interior?

No — alternate interior is on opposite sides.

But 135° is large — perhaps it's supplementary?

Wait — maybe the angle "?" is not alternate interior.

Wait — let's consider the adjacent angle.

At the bottom intersection, the 135° is interior, right.

Then the adjacent angle (on the same line) is 180° - 135° = 45°, which is exterior, on the right.

Now, the corresponding angle to that 45° would be on the top line, right, exterior.

But "?" is on the left, interior.

Wait — perhaps the vertical angle to the 135° is also 135°, but that’s not helpful.

Wait — another idea: the same-side interior angles are supplementary.

So:
- Bottom, interior, right = 135°
- Top, interior, right = ? → same-side interior → must add to 180° → so 45°

But "?" is on the left, not the right.

Wait — so the top, interior, left is alternate interior to the bottom, interior, right → so should be equal → 135°

But again, 135° not in options.

Unless I'm misreading the diagram.

Wait — maybe the 135° is exterior?

Look: the angle is labeled 135°, and it's on the bottom line, on the right, and the transversal is going down to the right.

If the angle is between the line and the transversal, and it's 135°, and it's outside, then the interior angle would be 180° - 135° = 45°.

Ah! That’s key.

So the 135° is exterior, on the right, bottom.

Then the interior angle at that point is 180° - 135° = 45°.

Now, the alternate interior angle to that 45° would be on the top line, left, interior — which is exactly where "?" is.

So "?" = 45°

✔ Answer: A) 45°

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17)

Given: 109° on the bottom line, on the right, interior.

Angle "?" is on the top line, on the left, interior.

So again, alternate interior angles → should be equal → 109°

But 109° is option D.

But wait — is it?

Wait — let’s confirm.

109° is interior, right, bottom.

Alternate interior → top, left, interior → yes.

So "?" = 109°

But is that correct?

Wait — could it be supplementary?

No — alternate interior are equal.

But let’s check: if the lines are parallel, then yes.

So "?" = 109°

✔ Answer: D) 109°

Wait — but let’s double-check.

Is the 109° interior? Yes — between the lines.

And "?" is also between the lines, on the opposite side.

So yes, alternate interior → equal.

So answer is D.

But let’s see the options: A) 131° B) 144° C) 93° D) 109°

Yes — D is there.

✔ Answer: D) 109°

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18)

There’s a right angle symbol (square) at the intersection of the transversal and the top line.

So the angle between the transversal and the top line is 90°.

Since the lines are parallel, and the transversal is perpendicular to the top line, it is also perpendicular to the bottom line.

So the angle at the bottom line, between the transversal and the line, is also 90°.

So "?" = 90°

✔ Answer: D) 90°

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19)

Given: 135° on the bottom line, on the right, interior.

Angle "?" is on the top line, on the left, interior.

So again, alternate interior angles → equal → 135°

But 135° is option A.

So is it?

Wait — is the 135° interior?

Yes — between the lines.

And "?" is on the opposite side, between the lines → alternate interior → equal.

So "?" = 135°

✔ Answer: A) 135°

Wait — but let’s confirm.

Could it be supplementary?

No — alternate interior are equal.

But wait — the 135° might be exterior?

Look: the angle is labeled 135°, and it's on the bottom, right, and the transversal is going down to the right.

If it's between the line and the transversal, and it's 135°, then the interior angle is 135° only if it's inside.

But 135° is greater than 90°, so it's obtuse — possible.

But let’s think: if the angle is 135°, then the adjacent angle (on the same line) is 45°.

But for alternate interior, if the 135° is interior, then "?" = 135°.

But let’s see the options: A) 135° B) 50° C) 105° D) 39°

So A is there.

But wait — maybe it's same-side interior?

No — same-side interior would be on the same side.

Here, "?" is on the left, 135° is on the right → opposite sides → alternate.

So yes, equal.

But wait — could it be that the 135° is exterior?

If the 135° is exterior, then the interior angle is 180° - 135° = 45°, and then alternate interior would be 45° — not in options.

But 45° is not listed.

Wait — unless it's supplementary.

Wait — let’s suppose the 135° is interior, then alternate interior = 135° → A

But let’s check: in problem 16, we had a similar situation, and we saw that if the angle was labeled outside, we used the interior.

But here, the 135° is likely interior, because it's shown between the lines.

Moreover, the diagram shows the angle opening toward the inside.

So yes, 135° is interior, so alternate interior = 135°

✔ Answer: A) 135°

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20)

Given: 133° on the top line, on the right, interior.

Angle "?" is on the bottom line, on the left, interior.

So again, alternate interior angles → equal → 133°

So "?" = 133°

✔ Answer: A) 133°

But wait — let’s confirm.

Top, interior, right = 133°

Bottom, interior, left = alternate interior → yes → 133°

Yes.

So all done.

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Final Answers:

13) C) 58°
14) C) 60°
15) D) 54°
16) A) 45°
17) D) 109°
18) D) 90°
19) A) 135°
20) A) 133°

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✔ Summary:

| Problem | Answer |
|--------|--------|
| 13 | C) 58° |
| 14 | C) 60° |
| 15 | D) 54° |
| 16 | A) 45° |
| 17 | D) 109° |
| 18 | D) 90° |
| 19 | A) 135° |
| 20 | A) 133° |

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