True/False worksheet on parallel lines cut by a transversal, featuring a diagram and questions about angle types and values.

Let’s go step by step through each True/False question using the diagram and angle relationships.

We are told:
- Lines L and M are parallel.
- Line N is a transversal cutting them.
- Angle 3 is given as 125°.
- Angles are labeled 1 to 8 around the two intersections.

Recall key angle relationships when parallel lines are cut by a transversal:

→ Vertical angles: opposite each other at an intersection → always equal
→ Corresponding angles: same relative position at each intersection → equal if lines are parallel
→ Alternate interior angles: inside the parallel lines, on opposite sides of transversal → equal
→ Alternate exterior angles: outside the parallel lines, on opposite sides of transversal → equal
→ Supplementary angles: add up to 180° (often adjacent angles on a straight line)

Also, angles on a straight line sum to 180°.

Given: ∠3 = 125°

At top intersection (line L and N):
- ∠3 and ∠4 are adjacent on a straight line → ∠4 = 180° - 125° = 55°
- ∠1 and 3 are vertical? Let’s check positions.

Looking at standard labeling:

Typically, for two lines intersecting:
Top-left: ∠1
Top-right: ∠2
Bottom-left: ∠3
Bottom-right: ∠4

But in this diagram, from the image description (even though we don’t describe it), we know:

Angle 3 is marked 125°, and it's below line L, left side of transversal N.

So likely:

At top intersection (L and N):
- Above line L, right of N: ∠1
- Below line L, left of N: ∠3 = 125°
- So ∠1 and ∠3 are NOT vertical — they are actually... let’s think.

Actually, vertical angles are directly across from each other.

If ∠3 is bottom-left at top intersection, then its vertical angle would be top-right — which is probably ∠1? Wait, no.

Standard numbering often goes:

For top intersection:
∠1 (top right)
∠2 (top left)
∠3 (bottom left)
∠4 (bottom right)

But in many diagrams like this, especially with transversals, the numbering might be:

From the problem statement and common usage:

Usually:
- ∠1 and 3 are NOT vertical — because vertical angles are opposite. If ∠3 is one angle, its vertical pair is the one directly across — which should be ∠1 only if they’re opposite.

Wait — let me reconstruct based on logic.

Since ∠3 = 125°, and it’s formed by line L and transversal N.

Angles around point where L and N meet:

Assume:
- ∠3 and ∠1 are vertical? Then they’d be equal. But let’s test that.

Actually, looking at typical textbook diagrams for “parallel lines cut by transversal” with 8 angles:

The usual labeling is:

Top intersection (line L and transversal N):
- Upper right: ∠1
- Upper left: ∠2
- Lower left: ∠3
- Lower right: ∠4

Bottom intersection (line M and transversal N):
- Upper right: ∠5
- Upper left: ∠6
- Lower left: ∠7
- Lower right: ∠8

Yes, that matches standard convention.

So:

At top intersection:
∠1 (upper right)
∠2 (upper left)
∠3 (lower left) = 125°
∠4 (lower right)

Then:
∠3 and 1 are NOT vertical — vertical to ∠3 is ∠1? No.

Vertical angles: ∠1 and ∠3 are not vertical — ∠1 is upper right, ∠3 is lower left — those are actually vertical! Yes!

In any two intersecting lines, the angles opposite each other are vertical.

So if you have two lines crossing, forming four angles:

Label them clockwise: A, B, C, D

Then A and C are vertical, B and D are vertical.

So if ∠3 is lower left, then the angle opposite (vertical) is upper right — which is ∠1.

Therefore, ∠1 and ∠3 ARE vertical angles → so they are equal.

But wait — ∠3 is given as 125°, so ∠1 should also be 125°? But let’s verify with adjacent angles.

∠3 and 4 are adjacent on line L → so ∠3 + 4 = 180° → ∠4 = 55°

Similarly, ∠1 and ∠2 are adjacent → ∠1 + ∠2 = 180°

And ∠1 and ∠3 are vertical → so ∠1 = 3 = 125°

Then ∠2 = 180° - 125° = 55°

Okay, that makes sense.

Now bottom intersection (M and N):

∠5 (upper right)
∠6 (upper left)
∠7 (lower left)
∠8 (lower right)

Because lines L and M are parallel, corresponding angles are equal.

Corresponding angles:
∠1 corresponds to ∠5 (both upper right) → so ∠5 = ∠1 = 125°? Wait no — earlier I said ∠1 = 125°, but let’s double-check.

Hold on — if ∠3 = 125°, and ∠3 is lower left at top, then:

Adjacent to ∠3 on the straight line L is ∠4 (lower right) → ∠4 = 180 - 125 = 55°

Vertical to ∠3 is ∠1 (upper right) → so ∠1 = 125°

Vertical to ∠4 is ∠2 (upper left) → ∠2 = 55°

Now, since L || M, corresponding angles:

∠1 (top, upper right) corresponds to ∠5 (bottom, upper right) → so ∠5 = ∠1 = 125°

∠2 (top, upper left) corresponds to ∠6 (bottom, upper left) → ∠6 = ∠2 = 55°

∠3 (top, lower left) corresponds to ∠7 (bottom, lower left) → ∠7 = 3 = 125°

∠4 (top, lower right) corresponds to ∠8 (bottom, lower right) → ∠8 = ∠4 = 55°

Also, alternate interior angles: between the parallels, opposite sides of transversal.

So ∠3 and ∠5? ∠3 is lower left top, ∠5 is upper right bottom — not alternate interior.

Alternate interior: ∠3 and ∠6? Let’s see:

Interior means between L and M.

So angles between L and M: ∠3, ∠4, ∠5, ∠6

Alternate interior: on opposite sides of transversal.

So ∠3 (left side, below L) and ∠5 (right side, above M)? Not quite.

Standard definition:

Alternate interior angles:
- One is on the left side of transversal, between the parallels
- The other is on the right side of transversal, between the parallels
- And they are not adjacent.

So for example: ∠3 and ∠5? ∠3 is below L, left of N; ∠5 is above M, right of N — yes, both between L and M, and on opposite sides of N → so they are alternate interior.

But ∠3 = 125°, ∠5 = 125° → so they are equal, which is correct for parallel lines.

Another pair: ∠4 and 6 → ∠4 = 55°, ∠6 = 55° → also alternate interior.

Alternate exterior: outside the parallels, opposite sides.

∠1 and 7: ∠1 is above L, right of N; ∠7 is below M, left of N → both exterior, opposite sides → alternate exterior → should be equal → ∠1=125°, ∠7=125° → yes.

Now let’s answer each question:

1. Angles 1 and 3 are vertical angles.

As established: ∠1 is upper right at top intersection, ∠3 is lower left at top intersection → they are opposite → vertical angles → TRUE.

2. The value of angle 7 is 125 degrees.

∠7 corresponds to ∠3 (since both are lower left at their respective intersections) → and since lines are parallel, corresponding angles equal → ∠7 = ∠3 = 125° → TRUE.

3. Angles 1 and 6 are alternate interior angles.

∠1 is above line L (exterior), ∠6 is between L and M (interior) → so one is exterior, one is interior → cannot be alternate interior. Alternate interior must both be interior. Also, ∠1 is right side, ∠6 is left side — but since ∠1 is not interior, this is false. FALSE.

4. Angles 3 and 5 are alternate interior angles.

∠3: between L and M? ∠3 is below L, so if L is top line, and M is bottom line, then "between" means above M and below L. ∠3 is below L, so yes, it's between if we consider the region. Actually, ∠3 is on the top line, but the angle itself is formed by the transversal and line L, so the angle ∠3 is located in the space between L and M? Not exactly — the vertex is on L, but the angle opens downward toward M.

In standard terms, angles 3,4,5,6 are considered the "interior" angles because they are between the two parallel lines.

∠3 is at top intersection, opening down → so it's interior.

∠5 is at bottom intersection, opening up → also interior.

And they are on opposite sides of the transversal: ∠3 is left, ∠5 is right → so yes, alternate interior → TRUE.

5. Angles 4 and 8 are corresponding angles.

∠4 is lower right at top intersection.

∠8 is lower right at bottom intersection.

Same relative position → corresponding angles → and since lines parallel, they are equal → ∠4=55°, ∠8=55° → so yes, corresponding → TRUE.

6. The value of angle 5 is 55 degrees.

Earlier we said ∠5 corresponds to ∠1, and ∠1=125° → so ∠5=125° → not 55° → FALSE.

7. Angles 1 and 7 are alternate exterior angles.

∠1: above L, right of N → exterior

∠7: below M, left of N → exterior

Opposite sides of transversal → alternate exterior → and they are equal (both 125°) → TRUE.

8. Angles 6 and 8 are corresponding angles.

∠6: upper left at bottom intersection

∠8: lower right at bottom intersection — same intersection! Corresponding angles are at different intersections.

Corresponding angles must be at different intersections, same relative position.

∠6 and ∠8 are at the same intersection (bottom), so they can't be corresponding. In fact, they are adjacent or something else.

∠6 and ∠8: at bottom intersection, ∠6 is upper left, ∠8 is lower right — they are vertical angles? Let's see: upper left and lower right are vertical → yes, so ∠6 and ∠8 are vertical angles → not corresponding.

Corresponding to ∠6 would be ∠2 (at top, upper left).

So FALSE.

9. Angles 1 and 6 are supplementary angles.

Supplementary means add to 180°.

∠1 = 125°, ∠6 = 55° → 125 + 55 = 180 → yes, supplementary → TRUE.

Why? Because ∠1 and 2 are supplementary (adjacent on straight line), and ∠2 = ∠6 (corresponding angles) → so ∠1 + ∠6 = 180°.

10. Angles 4 and 8 are supplementary angles.

∠4 = 55°, ∠8 = 55° → 55+55=110 ≠ 180 → not supplementary → FALSE.

They are corresponding and equal, but not supplementary.

Now, let's list all answers:

1. True
2. True
3. False
4. True
5. True
6. False
7. True
8. False
9. True
10. False

Double-checking:

Q1: ∠1 and 3 vertical → yes → T
Q2: ∠7 = ∠3 = 125° → T
Q3: ∠1 (exterior) and ∠6 (interior) — not both interior → F
Q4: ∠3 and 5 — both interior, opposite sides → alternate interior → T
Q5: ∠4 and ∠8 — both lower right → corresponding → T
Q6: ∠5 = 125°, not 55° → F
Q7: ∠1 and ∠7 — both exterior, opposite sides → alternate exterior → T
Q8: ∠6 and ∠8 — same intersection, vertical angles → not corresponding → F
Q9: ∠1=125°, ∠6=55° → sum 180 → supplementary → T
Q10: ∠4=55°, ∠8=55° → sum 110 → not supplementary → F

All seem correct.

Final Answer:
1. True
2. True
3. False
4. True
5. True
6. False
7. True
8. False
9. True
10. False