Geometry worksheet on parallel lines and transversals with angle classification and measurement problems.

Let's solve each problem step by step based on the image you provided. This is a geometry worksheet focusing on parallel lines and transversals, involving angle relationships such as:

- Vertical angles – congruent
- Corresponding angles – congruent when lines are parallel
- Alternate interior angles – congruent when lines are parallel
- Alternate exterior angles – congruent when lines are parallel
- Same-side interior (consecutive) angles – supplementary when lines are parallel
- Linear pairs – supplementary (add to 180°)

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Part 1: Classify each angle pair and indicate whether they are congruent or supplementary

We’ll refer to the diagram with two parallel lines $ w $ and $ y $, cut by a transversal.

Let’s label the angles as shown in the diagram:
- Top line: Angles 1, 2, 3, 4 (from left to right)
- Bottom line: Angles 5, 6, 7, 8 (from left to right)

Assuming standard labeling (clockwise from top-left):

```
w: 1 2
\ /
/ \
/ \
4 3

y: 5 6
\ /
/ \
/ \
8 7
```

But more likely, it's a standard configuration where:
- Lines $ w $ and $ y $ are parallel.
- Transversal intersects them forming 8 angles.

Standard positions:
- ∠1 and ∠5 are corresponding
- ∠2 and ∠6 are corresponding
- ∠3 and ∠7 are corresponding
- ∠4 and ∠8 are corresponding

Also:
- Vertical angles: ∠1 & ∠3, ∠2 & ∠4, ∠5 & ∠7, ∠6 & ∠8
- Linear pairs: e.g., ∠1 & ∠2, ∠1 & ∠4, etc.

Now let’s go through each:

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#### 1. ∠3 and ∠5
- These are alternate interior angles (between the parallel lines, on opposite sides of the transversal).
- Since $ w \parallel y $, they are congruent.

✔ Answer: Alternate Interior Angles → Congruent

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#### 2. ∠4 and ∠8
- These are corresponding angles (same position relative to the transversal and parallel lines).
- So, they are congruent.

✔ Answer: Corresponding Angles → Congruent

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#### 3. ∠2 and ∠5
- ∠2 is on the top line, outside; ∠5 is on the bottom line, inside.
- They are not corresponding, alternate, or same-side.
- Let’s check: ∠2 and ∠5 are not adjacent, not vertical, not alternate.
- Actually, ∠2 and ∠5 are not related directly unless we consider their relationship via other angles.

Wait — better to look at positions:
- ∠2 and ∠5: one is on top, one on bottom, both on the left side of transversal.
- But ∠2 is above, ∠5 is below — so they are not corresponding.
- Are they same-side interior? No — same-side interior would be like ∠3 and ∠5 (both inside, same side).
- ∠2 and ∠5 are not special pairs.
- Wait — actually, ∠2 and ∠5 are not a standard pair. But perhaps they are supplementary?

Wait — let's reconsider.

Actually, ∠2 and ∠5 are not a typical pair. But if we consider:
- ∠2 and ∠4 are vertical → congruent
- ∠4 and ∠8 are corresponding → congruent
- ∠5 and ∠7 are vertical

But ∠2 and ∠5: no direct relation.

Wait — maybe it's a typo? Or perhaps I need to re-express.

Alternatively, maybe ∠2 and ∠5 are not congruent or supplementary.

But let’s think: Is there any reason for them to be related?

No — unless the transversal forms a straight line.

Wait — actually, ∠2 and ∠5 are not adjacent or on a straight line.

So, no special relationship?

But wait — perhaps I misread.

Wait — let's assume the standard diagram.

Let me define clearly:

Typical labeling:
- Line $ w $: angles 1, 2, 3, 4 from top-left to top-right
- Line $ y $: angles 5, 6, 7, 8 from bottom-left to bottom-right

Then:
- ∠1 and ∠5: corresponding
- ∠2 and ∠6: corresponding
- ∠3 and ∠7: corresponding
- ∠4 and ∠8: corresponding

Vertical angles:
- ∠1 and ∠3
- ∠2 and ∠4
- ∠5 and ∠7
- ∠6 and ∠8

Now:

#### 3. ∠2 and ∠5
- ∠2 is on top line, right side?
Wait — confusion in labeling.

Better: Assume the transversal crosses from upper-left to lower-right.

Then:
- At top intersection: ∠1 (upper-left), ∠2 (upper-right), ∠3 (lower-right), ∠4 (lower-left)
- At bottom intersection: ∠5 (upper-left), ∠6 (upper-right), ∠7 (lower-right), ∠8 (lower-left)

So:
- ∠1 and ∠5: corresponding
- ∠2 and ∠6: corresponding
- ∠3 and ∠7: corresponding
- ∠4 and ∠8: corresponding

Vertical:
- ∠1 and ∠3
- ∠2 and ∠4
- ∠5 and ∠7
- ∠6 and ∠8

Now:

#### 3. ∠2 and ∠5
- ∠2 is upper-right on top line
- ∠5 is upper-left on bottom line
- Not corresponding, not alternate, not vertical
- But ∠2 and ∠5 are not adjacent or forming linear pair
- So no special relationship? That can't be.

Wait — maybe ∠2 and ∠5 are not a standard pair. But perhaps they are supplementary?

Let’s try to find if they form a linear pair or something.

No — they’re not adjacent.

Wait — perhaps the question is asking for relationship only if they are congruent or supplementary.

But in general, without specific values, we can’t say they are supplementary.

Wait — but in this case, since lines are parallel, maybe we can determine.

Actually, ∠2 and ∠5 are not necessarily congruent or supplementary.

But let’s see:

∠2 and ∠6 are corresponding → congruent
∠6 and ∠5 are adjacent on a straight line → supplementary

So ∠2 and ∠6 are congruent, and ∠6 + ∠5 = 180° → so ∠2 + ∠5 = 180° → supplementary

Yes!

Because:
- ∠2 ≅ ∠6 (corresponding)
- ∠6 and ∠5 are linear pair → supplementary → ∠6 + ∠5 = 180°
- So ∠2 + ∠5 = 180° → Supplementary

✔ So ∠2 and ∠5 are supplementary

→ Same-side exterior angles? Or just supplementary.

They are same-side exterior angles? Let's see: both are outside, on the same side of transversal.

Wait — ∠2 is on the right, ∠5 is on the left? No — depends.

If transversal goes from upper-left to lower-right, then:

- ∠2 is on the right side
- ∠5 is on the left side → different sides

So not same-side.

But anyway, we have: ∠2 ≅ ∠6, ∠6 + ∠5 = 180° → ∠2 + ∠5 = 180° → supplementary

So answer: Supplementary

✔ Answer: Supplementary

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#### 4. ∠2 and ∠8
- ∠2: upper-right on top
- ∠8: lower-left on bottom
- Are they related?

Note: ∠2 and ∠4 are vertical → congruent
∠4 and ∠8 are corresponding → congruent
So ∠2 ≅ ∠4 ≅ ∠8 → so ∠2 ≅ ∠8

So congruent

Are they alternate exterior? Let's see:
- ∠2 is upper-right → exterior
- ∠8 is lower-left → exterior
- On opposite sides of transversal? Yes
- And outside the lines → yes

So alternate exterior angles → congruent

✔ Answer: Alternate Exterior Angles → Congruent

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#### 5. ∠1 and ∠7
- ∠1: upper-left on top
- ∠7: lower-right on bottom
- Are they corresponding? No
- ∠1 and ∠5 are corresponding → ∠1 ≅ ∠5
- ∠5 and ∠7 are vertical → ∠5 ≅ ∠7
- So ∠1 ≅ ∠5 ≅ ∠7 → ∠1 ≅ ∠7

So congruent

What type? ∠1 and ∠7 are alternate interior angles?

Wait: ∠1 is upper-left, ∠7 is lower-right → both on opposite sides of transversal, and between the lines → yes, alternate interior angles

✔ Answer: Alternate Interior Angles → Congruent

---

#### 6. ∠2 and ∠6
- ∠2 and ∠6: both on the right side
- ∠2 on top, ∠6 on bottom → corresponding angles
- So congruent

✔ Answer: Corresponding Angles → Congruent

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✔ Summary for Part 1:

| Problem | Angle Pair | Relationship | Congruent or Supplementary |
|--------|------------|--------------|-----------------------------|
| 1 | ∠3 and ∠5 | Alternate Interior | Congruent |
| 2 | ∠4 and ∠8 | Corresponding | Congruent |
| 3 | ∠2 and ∠5 | Same-side exterior? But derived: ∠2 ≅ ∠6, ∠6 + ∠5 = 180° → ∠2 + ∠5 = 180° | Supplementary |
| 4 | ∠2 and ∠8 | Alternate Exterior | Congruent |
| 5 | ∠1 and ∠7 | Alternate Interior | Congruent |
| 6 | ∠2 and ∠6 | Corresponding | Congruent |

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Part 2: Find each angle measure

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#### 7. Given: $ w \parallel y $, $ m\angle 5 = 149^\circ $

We are given ∠5 = 149°

From diagram:
- ∠5 and ∠7 are vertical → ∠7 = 149°
- ∠5 and ∠6 are linear pair → ∠6 = 180° - 149° = 31°
- ∠6 and ∠2 are corresponding → ∠2 = 31°
- ∠2 and ∠4 are vertical → ∠4 = 31°
- ∠1 and ∠3 are vertical
- ∠1 and ∠5 are corresponding → ∠1 = 149°
- So ∠3 = 149°
- ∠8 and ∠6 are vertical → ∠8 = 31°
- ∠4 and ∠8 are corresponding → already consistent

Let’s fill in:

| m∠1 | m∠2 | m∠3 | m∠4 | m∠5 | m∠6 | m∠7 | m∠8 |
|-----|-----|-----|-----|-----|-----|-----|-----|
| 149° | 31° | 149° | 31° | 149° | 31° | 149° | 31° |

✔

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#### 8. Given: $ m \parallel n $, $ m\angle 1 = 74^\circ $

Wait — here the diagram has lines $ m $ and $ n $, and transversal $ p $. Angles labeled 1–8.

Given: ∠1 = 74°

Assume same labeling:
- ∠1 and ∠3 are vertical → ∠3 = 74°
- ∠1 and ∠5 are corresponding → ∠5 = 74°
- ∠5 and ∠7 are vertical → ∠7 = 74°
- ∠1 and ∠2 are linear pair → ∠2 = 180° - 74° = 106°
- ∠2 and ∠4 are vertical → ∠4 = 106°
- ∠4 and ∠6 are corresponding → ∠6 = 106°
- ∠6 and ∠8 are vertical → ∠8 = 106°

So:

| m∠1 | m∠2 | m∠3 | m∠4 | m∠5 | m∠6 | m∠7 | m∠8 |
|-----|-----|-----|-----|-----|-----|-----|-----|
| 74° | 106° | 74° | 106° | 74° | 106° | 74° | 106° |

✔

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#### 9. Given: $ a \parallel b $, $ m\angle 8 = 125^\circ $

Diagram: lines $ a $ and $ b $, transversal, angles 1–8.

Given: ∠8 = 125°

Assume standard:
- ∠8 and ∠6 are vertical → ∠6 = 125°
- ∠6 and ∠2 are corresponding → ∠2 = 125°
- ∠2 and ∠1 are linear pair → ∠1 = 180° - 125° = 55°
- ∠1 and ∠3 are vertical → ∠3 = 55°
- ∠3 and ∠7 are corresponding → ∠7 = 55°
- ∠7 and ∠5 are vertical → ∠5 = 55°
- ∠5 and ∠4 are linear pair → ∠4 = 180° - 55° = 125°
- ∠4 and ∠6 are vertical → already consistent

Wait: ∠4 and ∠6 are both 125° → okay

Now list:

| m∠1 | m∠2 | m∠3 | m∠4 | m∠5 | m∠6 | m∠7 | m∠8 |
|-----|-----|-----|-----|-----|-----|-----|-----|
| 55° | 125° | 55° | 125° | 55° | 125° | 55° | 125° |

✔

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#### 10. Given: $ x \parallel y $, $ m\angle 2 = 22^\circ $

Diagram: lines $ x $ and $ y $, transversal, angles 1–8.

Given: ∠2 = 22°

- ∠2 and ∠4 are vertical → ∠4 = 22°
- ∠2 and ∠6 are corresponding → ∠6 = 22°
- ∠6 and ∠8 are vertical → ∠8 = 22°
- ∠1 and ∠2 are linear pair → ∠1 = 180° - 22° = 158°
- ∠1 and ∠3 are vertical → ∠3 = 158°
- ∠3 and ∠7 are corresponding → ∠7 = 158°
- ∠7 and ∠5 are vertical → ∠5 = 158°
- ∠5 and ∠4 are linear pair → ∠5 + ∠4 = 158° + 22° = 180° → good

So:

| m∠1 | m∠2 | m∠3 | m∠4 | m∠5 | m∠6 | m∠7 | m∠8 |
|-----|-----|-----|-----|-----|-----|-----|-----|
| 158° | 22° | 158° | 22° | 158° | 22° | 158° | 22° |

✔

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#### 11. Given: $ r \parallel s $, $ m\angle 1 = 29^\circ $, $ m\angle 15 = 65^\circ $

This is a more complex diagram with multiple intersecting lines.

From the diagram (based on standard notation):
- Two parallel lines $ r $ and $ s $
- A transversal cuts them, forming angles 1–14
- But also, another line intersects at point O, forming angles 1–14 around a point

Wait — looks like two transversals intersecting at a point on the parallel lines.

Angles labeled 1–14, with some overlapping.

But given:
- $ r \parallel s $
- $ m\angle 1 = 29^\circ $
- $ m\angle 15 = 65^\circ $

Wait — angle 15? But only up to 14?

Possibly typo? Or maybe angle 15 is part of a different labeling.

Looking at the diagram:
- It shows two parallel lines $ r $ and $ s $
- Two transversals intersecting at a point, forming multiple angles
- Angles numbered 1 to 14 around the point

But angle 15 is mentioned — probably a mistake.

Wait — perhaps it's $ m\angle 1 = 29^\circ $, $ m\angle 14 = 65^\circ $? Or $ m\angle 15 $ is not present.

Alternatively, maybe angle 15 is a typo and should be angle 14 or 13.

But looking closely: the diagram shows angles 1 to 14, and angle 15 is not there.

Wait — perhaps it's $ m\angle 1 = 29^\circ $, $ m\angle 14 = 65^\circ $? But that doesn't make sense.

Wait — maybe the angle labeled "15" is actually an angle between the lines, but it's not numbered.

Alternatively, perhaps the diagram has two transversals crossing at a point, and angles are labeled 1–14 around the circle.

But then angle 15 doesn't exist.

Wait — maybe it's $ m\angle 1 = 29^\circ $, $ m\angle 15 = 65^\circ $ — but angle 15 is not in the diagram.

Alternatively, perhaps the number 15 is a typo and should be angle 14 or 13.

Wait — looking at the diagram: angles are labeled 1–14, and the last one is 14.

But the text says: “$ m\angle 15 = 65^\circ $” — that must be a typo.

Perhaps it's $ m\angle 14 = 65^\circ $? Or $ m\angle 13 $?

Wait — let’s assume it’s $ m\angle 1 = 29^\circ $, $ m\angle 14 = 65^\circ $, or perhaps $ m\angle 13 = 65^\circ $?

But without seeing the exact diagram, it’s hard.

But let’s assume the standard setup: two parallel lines $ r $ and $ s $, cut by two transversals intersecting at a point.

Let’s suppose:

- One transversal creates angles 1–8
- Another creates angles 9–14

But still, angle 15 is not present.

Alternatively, perhaps the angle labeled "15" is actually angle 14 or 13.

Wait — perhaps it’s $ m\angle 1 = 29^\circ $, $ m\angle 14 = 65^\circ $? But then we need to use parallel lines.

Alternatively, maybe angle 15 is a typo and should be angle 14.

But let’s assume the following:

In many such problems, when two lines intersect, vertical angles are equal.

And if $ r \parallel s $, then alternate interior angles are equal.

But without the diagram, it’s hard.

But let’s suppose:

- $ m\angle 1 = 29^\circ $
- $ m\angle 15 = 65^\circ $ — but 15 is not in diagram → likely typo

Wait — looking at the diagram again: angles are labeled 1–14, and angle 15 is not there.

But the text says: “$ m\angle 15 = 65^\circ $” — this must be a typo.

Possibility: it's $ m\angle 14 = 65^\circ $? Or $ m\angle 13 $? Or $ m\angle 10 $?

Alternatively, maybe it's $ m\angle 1 = 29^\circ $, and $ m\angle 15 $ is meant to be $ m\angle 14 $, or perhaps $ m\angle 15 $ is angle formed by intersection.

But since we can’t resolve it without the diagram, let’s assume a common configuration.

Wait — perhaps angle 15 is not present, and it's a typo for angle 14.

But let’s assume the following:

From the diagram, angle 1 is 29°, and angle 15 is not there.

Alternatively, perhaps the problem means $ m\angle 1 = 29^\circ $, $ m\angle 14 = 65^\circ $, and we need to find others.

But let’s skip for now.

Wait — perhaps the diagram shows a triangle or something else.

But based on standard problems, if $ r \parallel s $, and $ m\angle 1 = 29^\circ $, and $ m\angle 15 = 65^\circ $, but 15 isn’t there.

Wait — perhaps it’s $ m\angle 1 = 29^\circ $, $ m\angle 14 = 65^\circ $, and we use vertical/linear pairs.

But without knowing which angles are which, it’s hard.

Alternatively, maybe angle 15 is a typo and should be angle 14.

Let’s assume $ m\angle 1 = 29^\circ $, $ m\angle 14 = 65^\circ $

But even then, we need to know how they relate.

Wait — perhaps angle 1 and angle 14 are on the same side.

But let’s give up and assume the diagram is standard.

Alternatively, perhaps the angle labeled "15" is actually angle 14, and it's 65°.

But since I can’t resolve it, I’ll leave it for now.

But let’s assume:

- $ m\angle 1 = 29^\circ $
- $ m\angle 15 = 65^\circ $ → likely typo, perhaps $ m\angle 14 = 65^\circ $

But without the diagram, I can’t proceed accurately.

However, in many such problems, when two lines intersect, vertical angles are equal, and parallel lines give congruent alternate interior angles.

But due to lack of clarity, I’ll stop here.

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✔ Final Answers:

#### Part 1: Classification

| # | Angle Pair | Relationship | Congruent or Supplementary |
|---|------------|--------------|----------------------------|
| 1 | ∠3 and ∠5 | Alternate Interior | Congruent |
| 2 | ∠4 and ∠8 | Corresponding | Congruent |
| 3 | ∠2 and ∠5 | Supplementary (via ∠6) | Supplementary |
| 4 | ∠2 and ∠8 | Alternate Exterior | Congruent |
| 5 | ∠1 and ∠7 | Alternate Interior | Congruent |
| 6 | ∠2 and ∠6 | Corresponding | Congruent |

#### Part 2: Angle Measures

7. Given $ m\angle 5 = 149^\circ $

| m∠1 | m∠2 | m∠3 | m∠4 | m∠5 | m∠6 | m∠7 | m∠8 |
|-----|-----|-----|-----|-----|-----|-----|-----|
| 149° | 31° | 149° | 31° | 149° | 31° | 149° | 31° |

8. Given $ m\angle 1 = 74^\circ $

| m∠1 | m∠2 | m∠3 | m∠4 | m∠5 | m∠6 | m∠7 | m∠8 |
|-----|-----|-----|-----|-----|-----|-----|-----|
| 74° | 106° | 74° | 106° | 74° | 106° | 74° | 106° |

9. Given $ m\angle 8 = 125^\circ $

| m∠1 | m∠2 | m∠3 | m∠4 | m∠5 | m∠6 | m∠7 | m∠8 |
|-----|-----|-----|-----|-----|-----|-----|-----|
| 55° | 125° | 55° | 125° | 55° | 125° | 55° | 125° |

10. Given $ m\angle 2 = 22^\circ $

| m∠1 | m∠2 | m∠3 | m∠4 | m∠5 | m∠6 | m∠7 | m∠8 |
|-----|-----|-----|-----|-----|-----|-----|-----|
| 158° | 22° | 158° | 22° | 158° | 22° | 158° | 22° |

11. Given $ m\angle 1 = 29^\circ $, $ m\angle 15 = 65^\circ $ — Incomplete due to unclear diagram. Likely typo in angle number.

Please double-check the diagram for angle 15.

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Let me know if you can clarify the diagram for problem 11!