Parallel Lines Cut by a Transversal Worksheets - Free Printable

Let’s solve each problem step by step. The key idea is: When two parallel lines are cut by a transversal, alternate interior angles are equal. Also, vertical angles are equal, and corresponding angles are equal — but here we’re focusing on *alternate* angles (which usually means alternate interior or alternate exterior).

We’ll go one figure at a time.

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Figure 1:
Two horizontal parallel lines, cut by a diagonal transversal.
Angle labeled “x” is on the top line, right side of transversal.
Angle labeled “50°” is on the bottom line, left side of transversal → these are alternate interior angles.
→ So x = 50°

✔ Answer for 1: x = 50

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Figure 2:
Two vertical parallel lines, cut by a diagonal transversal.
Angle labeled “x” is on the left line, top-right corner.
Angle labeled “70°” is on the right line, bottom-left corner → these are alternate interior angles.
→ So x = 70°

✔ Answer for 2: x = 70

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Figure 3:
Two horizontal parallel lines, cut by a diagonal transversal.
Angle labeled “x” is on the top line, left side.
Angle labeled “110°” is on the bottom line, right side → these are alternate exterior angles? Wait — actually, let’s check positions.

Actually, looking carefully:
The 110° angle is on the bottom line, right side, above the line? No — in standard diagrams, if it's between the parallels, it’s interior. But here, since both angles are on opposite sides of the transversal and outside the parallel lines? Let me think again.

Wait — better approach: use vertical angles + alternate interior.

In Figure 3:
- The 110° angle is on the bottom line, right side, below the line? Actually, no — typically in such diagrams, the angle shown is the one formed between the transversal and the line.

But note: if you look at the angle adjacent to 110° on the same straight line, it would be 70° (since 180 - 110 = 70). That 70° angle is an alternate interior angle with x.

So:
Adjacent angle to 110° = 180° - 110° = 70°
That 70° is alternate interior to x → so x = 70°

Alternatively, if 110° and x are alternate exterior, they should be equal — but that doesn’t match unless... wait, maybe I misread.

Actually, let’s label:

Top line: angle x is on the left, above the transversal? Or below?

Standard convention: In most textbooks, when they show an angle like this without specifying, it’s the acute or obtuse angle as drawn.

Looking at typical problems: If 110° is given on the bottom right, and x is on the top left, and lines are parallel, then x and 110° are corresponding angles? No — corresponding would be same relative position.

Actually, x and 110° are alternate exterior angles — which are also equal when lines are parallel.

Yes! Alternate exterior angles are equal.

So if 110° is on the bottom right (exterior), and x is on the top left (also exterior, opposite side of transversal) → they are alternate exterior → equal.

So x = 110°

Wait — but that contradicts my earlier thought. Let me double-check with logic.

If two parallel lines are cut by a transversal:

- Alternate interior angles: inside the parallels, opposite sides of transversal → equal
- Alternate exterior angles: outside the parallels, opposite sides of transversal → also equal

In Figure 3:
- The 110° angle is likely the one on the bottom line, on the right side, and it’s the angle *outside* the parallel lines? Or inside?

Actually, in many diagrams, if the angle is marked between the transversal and the line, and it’s on the "outer" part, it’s exterior.

But let’s assume standard positioning:

Typically, in such grids, for Figure 3:

Top line: angle x is on the upper left, formed by transversal going down to the right.

Bottom line: angle 110° is on the lower right, formed by the same transversal.

These are alternate exterior angles → so they are equal → x = 110°

But wait — sometimes students get confused. Let’s use another method.

Draw imaginary: if you slide the top angle down along the transversal, does it match the bottom angle? For alternate exterior, yes, they should be equal.

I recall that in some curricula, “alternate angles” refers specifically to alternate interior, but the title says “Alternate Angles in Parallel Lines”, which includes both.

But let’s check Figure 4 to see pattern.

Perhaps I should calculate based on supplementary angles.

Another way: the angle vertically opposite to 110° is also 110°, and that might be corresponding to x? Not necessarily.

Let’s do this systematically.

For Figure 3:

Assume the transversal cuts the two parallel lines.

At the bottom intersection: one angle is 110°. Since it’s a straight line, the adjacent angle (on the same side) is 70°.

Now, that 70° angle is on the bottom line, left side of transversal.

Then, the alternate interior angle to that 70° would be on the top line, right side of transversal.

But x is on the top line, left side — so not directly alternate interior.

x and the 70° angle are on the same side of the transversal, both above their respective lines? This is messy.

Better: use the fact that consecutive interior angles are supplementary.

Or just accept that in standard problems like this, if 110° is given and x is on the other side, and it's alternate, it's often 110°.

But let's look at Figure 4 for clue.

Figure 4:
Two horizontal parallel lines, transversal.
Angle x on top line, right side.
Angle 60° on bottom line, left side → alternate interior → x = 60°

Similarly, Figure 5:
Vertical parallel lines, transversal.
x on left line, bottom-left.
40° on right line, top-right → alternate interior → x = 40°

Figure 6:
Horizontal parallels, transversal.
x on top line, left side.
120° on bottom line, right side → alternate exterior → x = 120°? Or is it supplementary?

Wait, 120° and its adjacent angle is 60°, and if x is alternate interior to that 60°, then x=60°.

This is confusing. Let me define clearly.

General rule: When two parallel lines are cut by a transversal:

- Corresponding angles are equal.
- Alternate interior angles are equal.
- Alternate exterior angles are equal.
- Consecutive interior angles are supplementary (add to 180°).

In Figure 3:
Suppose the 110° angle is the one on the bottom line, on the right side, and it's the angle between the transversal and the line, measured from the line up to the transversal. If the transversal is going from top-left to bottom-right, then at the bottom intersection, the angle on the right side could be the one below the line or above.

In most textbook diagrams, the angle marked is the one inside the "V" formed, so for bottom line, if transversal comes from top-left, the angle on the right side of transversal at bottom line is the one between the transversal and the extension of the line to the right — which would be the exterior angle if we consider the region between the parallels as interior.

To avoid confusion, let's use this trick:

For any such diagram, find the angle that is either corresponding or alternate to x, and set equal.

In Figure 3:
The angle that is vertically opposite to the 110° angle is also 110°. That vertically opposite angle is on the bottom line, left side, above the line? No.

Vertically opposite means directly across the intersection point.

So at the bottom intersection, if one angle is 110°, the vertically opposite angle is also 110°, and it's on the other side of the transversal.

Then, that 110° angle (vertically opposite) is on the bottom line, left side, and it's an alternate interior angle to x? Let's see:

If x is on the top line, left side, and the 110° vertically opposite is on the bottom line, left side — that would be corresponding angles, not alternate.

Corresponding angles are in the same relative position at each intersection.

So if x is top-left, and the angle at bottom-left is 110°, then they are corresponding → so x = 110°.

Yes! That makes sense.

In Figure 3:
- At bottom intersection, the angle given as 110° is probably the one on the right side, but its vertically opposite angle is on the left side, and that left-side angle is corresponding to x (both on the left side of transversal, and both above their respective lines? Or both below?).

Actually, if the transversal is slanting down to the right, then at the top intersection, the angle on the left side of transversal and above the top line is x.

At the bottom intersection, the angle on the left side of transversal and below the bottom line is the vertically opposite to the 110° if 110° is on the right side below.

This is too ambiguous.

Let me assign coordinates mentally.

Assume for Figure 3:
- Top line: y = 1
- Bottom line: y = 0
- Transversal: line from (0,1) to (1,0), so slope -1.

At top intersection (0,1): the angle between the transversal and the top line. The top line is horizontal. The transversal has direction vector (1,-1), so the angle with positive x-axis is 315° or -45°, so the acute angle with the line is 45°, but that's not helping.

Perhaps in the diagram, the 110° is the obtuse angle at the bottom right.

Then, the alternate exterior angle to it would be at the top left, which is x, and since alternate exterior are equal, x = 110°.

I think that's standard.

Moreover, in many online sources, for similar figures, if it's alternate and the angle is given as 110°, x is 110°.

Let's move to Figure 6 to verify.

Figure 6:
x on top line, left side.
120° on bottom line, right side.

If they are alternate exterior, x = 120°.

But let's calculate: the adjacent angle to 120° is 60°, and if that 60° is alternate interior to x, then x=60°.

Which is it?

I recall that in some systems, "alternate angles" means alternate interior only, but the title says "Alternate Angles", which can include exterior.

But to resolve, let's look at the answer patterns.

Perhaps for consistency, in all cases where the angles are on opposite sides of the transversal and both outside or both inside, they are equal.

For Figure 3: 110° and x are both "outside" the parallel lines if we consider the space between them as inside, and on opposite sides of transversal → alternate exterior → equal → x=110°.

For Figure 6: 120° and x are similarly placed → x=120°.

But let's check Figure 9.

Figure 9:
Two vertical parallel lines, transversal.
x on left line, bottom-left.
50° on right line, top-right → alternate interior → x=50°.

Figure 10:
Vertical parallels, transversal.
x on left line, top-left.
70° on right line, bottom-right → alternate interior → x=70°.

Now back to Figure 3 and 6.

Another way: in Figure 3, if x and 110° were supplementary, that would be if they were consecutive interior, but they are not on the same side.

They are on opposite sides, so likely equal.

I found a better way: in the diagram, the angle marked 110° is likely the one that is not the acute one, and x is on the other side, so for alternate exterior, they are equal.

Moreover, in many homework sheets, for such a setup, x = 110°.

Similarly for Figure 6, x = 120°.

But let's confirm with Figure 4: x and 60° are alternate interior, so x=60° — that's clear.

Figure 5: x and 40° are alternate interior, x=40°.

Figure 7:
Two horizontal parallels, transversal.
x on top line, right side.
30° on bottom line, left side → alternate interior → x=30°.

Figure 8:
Horizontal parallels, transversal.
x on top line, left side.
100° on bottom line, right side → alternate exterior → x=100°.

Yes, that seems consistent.

So for Figure 3: x = 110°

Figure 6: x = 120°

Figure 8: x = 100°

Now let's list all:

1. x = 50° (alternate interior)
2. x = 70° (alternate interior)
3. x = 110° (alternate exterior)
4. x = 60° (alternate interior)
5. x = 40° (alternate interior)
6. x = 120° (alternate exterior)
7. x = 30° (alternate interior)
8. x = 100° (alternate exterior)
9. x = 50° (alternate interior)
10. x = 70° (alternate interior)

All seem consistent.

Just to be thorough, let's verify Figure 3 again.

In Figure 3: if the 110° is at the bottom right, and it's the angle between the transversal and the bottom line, measured clockwise or counterclockwise. If the transversal is going down to the right, then at the bottom intersection, the angle on the right side of the transversal and below the bottom line is 110°. Then, the alternate exterior angle would be at the top intersection, on the left side of the transversal and above the top line, which is x. And since alternate exterior angles are equal, x = 110°.

Similarly for others.

So final answers:

1. 50
2. 70
3. 110
4. 60
5. 40
6. 120
7. 30
8. 100
9. 50
10. 70

Final Answer:
1. 50
2. 70
3. 110
4. 60
5. 40
6. 120
7. 30
8. 100
9. 50
10. 70