Mathematics worksheet on parallel lines cut by a transversal, including angle congruence and measurement exercises.

Let’s solve this step by step.

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Test I: Parallel Lines cut by a Transversal — Congruent (C) or Supplementary (S)?

We are told that line l₁ is parallel to line l₂, and a transversal cuts them. We need to say whether each pair of angles is congruent (same measure) or supplementary (add up to 180°).

Recall:
- Congruent angles: same size (like corresponding angles, alternate interior/exterior, vertical angles)
- Supplementary angles: add to 180° (like consecutive interior/exterior angles, linear pairs)

Looking at the diagram:

Angles at top intersection (l₁):
Top-left = ∠8, top-right = ∠7
Bottom-left = ∠5, bottom-right = ∠6

Angles at bottom intersection (l₂):
Top-left = ∠4, top-right = ∠3
Bottom-left = ∠1, bottom-right = ∠2

Now go one by one:

1. ∠3 and ∠5 → These are *alternate interior angles* → Congruent (C)
2. ∠2 and ∠6 → These are *corresponding angles* → Congruent (C)
3. ∠4 and ∠5 → These are *consecutive interior angles* → Supplementary (S)
4. ∠1 and ∠7 → These are *alternate exterior angles* → Congruent (C)
5. ∠2 and ∠7 → These are *consecutive exterior angles* → Supplementary (S)
6. ∠4 and ∠6 → These are *vertical angles*? Wait — no, they’re on different intersections. Actually, ∠4 and 6 are *alternate exterior angles*? Let’s check: ∠4 is top-left on bottom line, ∠6 is bottom-right on top line → actually, they are *alternate exterior angles* → Congruent (C)
Wait — let me double-check:
Actually, ∠4 and ∠6:
∠4 is above l₂, left of transversal
∠6 is below l₁, right of transversal → Not standard pair.
Better way: Use positions.

Actually, better to use known rules:

Standard pairs:

- Corresponding: same position at each intersection → e.g., ∠1 & ∠5, ∠2 & ∠6, ∠3 & ∠7, ∠4 & ∠8 → all congruent
- Alternate Interior: between lines, opposite sides of transversal → ∠3 & ∠5, ∠4 & ∠6 → congruent
- Alternate Exterior: outside lines, opposite sides → ∠1 & ∠7, ∠2 & ∠8 → congruent
- Consecutive Interior (same side interior): ∠3 & ∠6, 4 & ∠5 → supplementary
- Consecutive Exterior: ∠1 & ∠8, ∠2 & ∠7 → supplementary

So let’s redo with this:

1. ∠3 and ∠5 → alternate interior → C
2. ∠2 and ∠6 → corresponding → C
3. ∠4 and ∠5 → consecutive interior → S
4. ∠1 and ∠7 → alternate exterior → C
5. ∠2 and ∠7 → consecutive exterior → S
6. ∠4 and ∠6 → alternate interior → C
7. ∠1 and 5 → corresponding → C
8. ∠3 and ∠6 → consecutive interior → S
9. ∠2 and ∠8 → alternate exterior → C
10. ∠1 and ∠8 → consecutive exterior → S

✔ All checked.

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Test II: Given m // n, find measures.

We’re told one angle is 70° — it’s labeled near ∠2 and ∠7? Looking at diagram:

At top intersection (line m), the angle marked 70° is adjacent to ∠1 and ∠2. Specifically, it’s the angle between the transversal and line m on the right side — so that’s ∠2? Or is it labeled next to ∠2?

Actually, looking: “70°” is written next to the angle that is vertically opposite to ∠7? Or adjacent?

In the diagram:
At top intersection:
Left side: ∠1 (top-left), ∠7 (bottom-left)
Right side: 70° (which must be ∠2, since ∠2 is bottom-right), and ∠? top-right is unlabeled but should be ∠? Wait — standard labeling:

Usually, for two lines cut by transversal:

Top line:
Top-left = ∠1, top-right = ∠2
Bottom-left = ∠7, bottom-right = ∠8? But here it’s labeled differently.

Wait — in Test II diagram:

It says:
At top intersection (line m):
∠1 is top-left
Then 70° is written near the angle that is bottom-right → which is ∠2
Then ∠7 is bottom-left
And top-right is unlabeled? But probably ∠? — wait, no, in the list we have ∠1, ∠2, ∠7 — so likely:

Assume:
At top intersection:
- Top-left: ∠1
- Bottom-right: ∠2 (and it’s marked 70°? Or is 70° the angle adjacent to ∠2?)

Actually, looking again: “70°” is written inside the angle that is between the transversal and line m on the right side — that would be ∠2 if ∠2 is bottom-right.

But in standard notation, when they write “70°” next to an angle, it means that angle is 70°.

In the diagram, it’s placed where ∠2 is — so let’s assume m∠2 = 70°

Then:

Since lines m and n are parallel, and t is transversal.

First, at top intersection:

∠1 and ∠2 are adjacent on a straight line → so they are supplementary → m∠1 + m∠2 = 180°
→ m∠1 = 180° - 70° = 110°

∠7 and ∠2 are vertical angles? No — ∠7 is bottom-left, ∠2 is bottom-right — they are adjacent on the same side? Actually, ∠7 and ∠2 are on the same side of the transversal, but different lines? No — same intersection.

At one intersection, angles around point:

∠1 and ∠7 are adjacent (left side), ∠1 and the top-right angle are adjacent, etc.

Actually, ∠1 and ∠7 are vertical? No — vertical angles are opposite.

Standard: at intersection, vertical angles are equal.

So:

- ∠1 and the angle opposite to it (which would be ∠2? No — opposite of ∠1 is the angle diagonally across — which in this case, if ∠1 is top-left, then opposite is bottom-right → which is ∠2? But ∠2 is labeled 70°, and ∠1 is 110° — not equal.

I think I misassigned.

Let me redefine based on common labeling:

In many textbooks, for two parallel lines cut by transversal:

Label the top intersection angles as:

Top-left: ∠1
Top-right: ∠2
Bottom-left: ∠3
Bottom-right: ∠4

But here, in Test II, it's labeled:

At top: ∠1 (left), and 70° is shown on the right side — probably meaning the angle on the right is 70°, which could be ∠2 if ∠2 is top-right.

But in the list, we have ∠1, ∠2, ∠7 — so likely:

Perhaps:

Top intersection:
- Left side: ∠1 (above line m), ∠7 (below line m)
- Right side: let’s say ∠? (above), ∠2 (below) — and the 70° is written next to the angle that is below line m on the right — so that’s ∠2.

So assume: m∠2 = 70°

Then:

- ∠1 and ∠2 are adjacent angles on a straight line (along line m) → so m∠1 + m∠2 = 180° → m∠1 = 110°

- ∠7 and ∠2 are vertical angles? No — ∠7 is below line m on left, ∠2 is below line m on right — they are not vertical. Vertical to ∠2 would be the angle above line m on left — which is ∠1? No.

Actually, at the top intersection:

The four angles are:

- Top-left: let's call it A
- Top-right: B
- Bottom-left: C
- Bottom-right: D

In the diagram, ∠1 is probably A (top-left), ∠7 is C (bottom-left), and ∠2 is D (bottom-right), and the 70° is labeled at D, so m∠2 = 70°.

Then:

- ∠1 and ∠2 are not adjacent; ∠1 and the top-right angle are adjacent.

Actually, ∠1 and 7 are adjacent (both on left side), and they form a linear pair with the angles on the right.

Better: angles on a straight line sum to 180°.

So along line m: the angles above and below on the left side: ∠1 and ∠7 are adjacent and form a straight line? No — ∠1 is above, ∠7 is below, but they are on the same side of the transversal — actually, they are adjacent angles sharing the transversal ray.

Standard: at intersection, any two adjacent angles sum to 180°.

So:

- ∠1 and the angle to its right (top-right) sum to 180°
- ∠1 and ∠7 sum to 180°? Only if they are on a straight line — but they are on different rays.

Actually, ∠1 and ∠7 are vertical angles? No.

Let’s think differently.

From the diagram description, since 70° is marked, and it's likely the measure of the angle shown, and from context, it's probably ∠2 = 70°.

Then, since ∠1 and ∠2 are on a straight line (along the transversal? No — along line m).

Actually, ∠1 and 2 are not on the same line. Let's list the relationships.

Assume:

At top intersection (line m and transversal t):

- The angle marked 70° is the one that is vertically opposite to ∠7? Or adjacent to ∠1?

Perhaps the 70° is the measure of the angle that is not labeled with a number — but in the text, it's written as "70°" next to the angle, and in the list, we have to find ∠1, ∠2, etc.

To resolve, let's look at the bottom part.

At bottom intersection (line n and transversal t):

Angles: ∠6 (top-left), ∠3 (top-right), ∠5 (bottom-left), ∠4 (bottom-right)

And we know that corresponding angles are equal.

If we can find one, we can find others.

But we have 70° at top.

Let me assume that the 70° is the measure of ∠2, as it's commonly placed.

So m∠2 = 70°

Then:

- ∠1 and ∠2 are adjacent angles that form a linear pair along the transversal? No — along line m.

Actually, ∠1 and the angle directly across from it are vertical.

Standard rule: vertical angles are equal.

So at top intersection:

- ∠1 and the angle opposite to it (which would be the bottom-right angle, i.e., ∠2) are vertical angles? Only if they are opposite.

If ∠1 is top-left, then vertical angle is bottom-right, which is ∠2.

But if m∠2 = 70°, then m∠1 = 70°? But earlier I thought they are supplementary.

I think there's confusion in labeling.

Let me look back at the image description.

In the user's image, for Test II, it says:

"Given: m // n"

Diagram:

Transversal t cutting lines m and n.

At top intersection (m):

- Left side: ∠1 (above m), ∠7 (below m)
- Right side: 70° is written, and it's likely the angle below m on the right, which is ∠2

So probably, the angle labeled 70° is ∠2, so m∠2 = 70°

Then, at the same intersection, ∠1 and ∠2 are not vertical; rather, ∠1 and the angle above m on the right are adjacent.

Actually, ∠1 and ∠7 are on the left, and they are adjacent angles that sum to 180° because they form a straight line along the transversal? No.

Let's define:

The four angles at top intersection:

- Angle between m and t, top-left: this is ∠1
- Angle between m and t, top-right: let's call it X
- Angle between m and t, bottom-left: ∠7
- Angle between m and t, bottom-right: ∠2 = 70°

Then, ∠1 and X are adjacent on line m, so m∠1 + mX = 180°
Similarly, ∠7 and ∠2 are adjacent on line m, so m∠7 + m∠2 = 180°
Also, ∠1 and ∠7 are adjacent on transversal t, so m∠1 + m∠7 = 180°? No, only if they are on a straight line, but they are on different rays.

Actually, at a point, the sum of angles around a point is 360°, and adjacent angles on a straight line sum to 180°.

So, along line m: the angles on one side of the transversal sum to 180°.

Specifically, the angle above m on left (∠1) and the angle above m on right (X) are adjacent and sum to 180° because they form a straight line along m.

Similarly, the angle below m on left (∠7) and below m on right (∠2) sum to 180°.

Also, vertically opposite angles are equal: so ∠1 = ∠2? No, vertical to ∠1 is the angle below m on right, which is ∠2, so if vertical angles are equal, then m∠1 = m∠2.

But if m∠2 = 70°, then m∠1 = 70°, but then along line m, ∠1 + X = 180°, so X = 110°, and similarly, ∠7 + ∠2 = 180°, so ∠7 = 110°, and vertical to ∠7 is X, so X = 110°, consistent.

But in the diagram, it's likely that the 70° is not ∠2, but the angle that is not labeled, or perhaps it's ∠7 or something.

Perhaps the 70° is the measure of the angle that is corresponding to something.

Another way: in many such problems, the 70° is given as the measure of the acute angle, and we use that.

Let's assume that the angle marked 70° is the one that is vertically opposite to ∠7 or something.

To avoid confusion, let's use the fact that when two parallel lines are cut by a transversal, corresponding angles are equal, alternate interior are equal, etc.

From the diagram, the 70° is likely the measure of the angle at the top right, which is not labeled with a number, but in the list, we have to find ∠1, ∠2, etc.

Perhaps ∠2 is the angle at bottom right of top intersection, and 70° is given for the angle at top right.

Let's read the diagram description again: "70°" is written next to the angle that is between the transversal and line m on the right side — and in standard labeling, if ∠1 is top-left, then the top-right angle might be unlabeled, but in the list, we have ∠1, ∠2, ∠7, so probably ∠2 is the bottom-right angle.

Perhaps the 70° is the measure of ∠2.

I recall that in some diagrams, the angle is labeled with the number and the measure.

To resolve, let's look at the bottom part.

At bottom intersection, we have ∠3, ∠4, 5, ∠6.

And we know that if we can find one, we can find others.

But we have only one given: 70° at top.

Another idea: the 70° is the measure of the angle that is corresponding to ∠3 or something.

Perhaps the 70° is the measure of the angle at the top, and it's the same as the corresponding angle at the bottom.

Let's assume that the 70° is the measure of the angle that is in the position of ∠2 at the top, but since it's not labeled, perhaps it's the angle that is vertically opposite to ∠7.

I think I need to make a decision.

Let me assume that the angle marked 70° is the measure of ∠2, as it's commonly done.

So m∠2 = 70°

Then, since ∠1 and ∠2 are vertical angles? No, in standard, if ∠1 is top-left, ∠2 is bottom-right, then they are vertical angles, so m∠1 = m∠2 = 70°

But then, along the line, the adjacent angles should sum to 180°.

For example, ∠1 and the top-right angle sum to 180°, so top-right = 110°

Similarly, ∠7 and ∠2 sum to 180°, so m∠7 = 110°

And vertical to ∠7 is the top-right angle, which is 110°, good.

Then for the bottom intersection, since m // n, corresponding angles are equal.

Corresponding to ∠1 (top-left) is ∠6 (top-left at bottom) , so m∠6 = m∠1 = 70°

Corresponding to ∠2 (bottom-right at top) is ∠4 (bottom-right at bottom) , so m∠4 = m∠2 = 70°

Corresponding to ∠7 (bottom-left at top) is ∠5 (bottom-left at bottom) , so m∠5 = m∠7 = 110°

Corresponding to the top-right angle (110°) is ∠3 (top-right at bottom) , so m∠3 = 110°

Now, let's verify with alternate interior angles.

Alternate interior: ∠3 and ∠7 should be equal? ∠3 is top-right at bottom, ∠7 is bottom-left at top — not alternate interior.

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

So for example, ∠3 and ∠7: ∠3 is above line n, right of transversal; ∠7 is below line m, left of transversal — not the same region.

Better: alternate interior angles are like ∠3 and ∠7? No.

Standard: alternate interior angles are ∠3 and 7? Let's see:

Typically, for two lines, alternate interior are the ones inside the parallel lines, on opposite sides of transversal.

So at top, the interior angles are ∠7 and ∠2 (below line m)
At bottom, interior angles are ∠6 and ∠3 (above line n)

So alternate interior would be ∠7 and ∠3, or ∠2 and ∠6.

Yes! So ∠7 and ∠3 are alternate interior angles, so they should be equal.

If m∠7 = 110°, then m∠3 = 110°, which matches what I have.

Similarly, ∠2 and ∠6 are alternate interior, m∠2 = 70°, m∠6 = 70°, good.

Also, consecutive interior: ∠7 and ∠6 should be supplementary, 110° + 70° = 180°, good.

So everything checks out.

Therefore:

m∠1 = 70° (vertical to ∠2)
m∠2 = 70° (given)
m∠3 = 110° (alternate interior to ∠7, or corresponding to top-right)
m∠4 = 70° (corresponding to ∠2)
m∠5 = 110° (corresponding to ∠7)
m∠6 = 70° (corresponding to ∠1)
m∠7 = 110° (adjacent to ∠2 on line m, so 180° - 70° = 110°)

Now for the sums:

18. m∠7 + m∠2 = 110° + 70° = 180°
19. m∠2 + m∠3 = 70° + 110° = 180°
20. m∠1 + m∠5 = 70° + 110° = 180°

All make sense.

So final answers for Test II:

11. m∠1 = 70°
12. m∠2 = 70°
13. m∠3 = 110°
14. m∠4 = 70°
15. m∠5 = 110°
16. m∠6 = 70°
17. m∠7 = 110°
18. m∠7 + m∠2 = 180°
19. m∠2 + m∠3 = 180°
20. m∠1 + m∠5 = 180°

Now, to confirm, in the diagram, is ∠2 really 70°? In my assumption, I set m∠2 = 70°, and it worked consistently.

If the 70° was meant to be another angle, it might not work, but given the context, this is logical.

So I'll go with that.

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Final Answers:

Test I:
1. C
2. C
3. S
4. C
5. S
6. C
7. C
8. S
9. C
10. S

Test II:
11. 70°
12. 70°
13. 110°
14. 70°
15. 110°
16. 70°
17. 110°
18. 180°
19. 180°
20. 180°

Now, box the final answer as per instruction.

Since the response should have only the final answer after reasoning, and the reasoning is above, I'll write the final answer clearly.

Note: For Test I, the answer is a sequence of C and S.

For Test II, it's numerical values.

So:

Final Answer:
Test I: 1.C 2.C 3.S 4.C 5.S 6.C 7.C 8.S 9.C 10.S
Test II: 11.70° 12.70° 13.110° 14.70° 15.110° 16.70° 17.110° 18.180° 19.180° 20.180°