Eighth Grade Parallel Lines and Transversals Activity - Free Printable

Let’s go step by step to solve each question using the diagram of parallel lines X and Y cut by transversal Z.

We’ll use basic angle relationships:
- Corresponding angles are in matching corners (like top-left, bottom-right) — they’re equal if lines are parallel.
- Alternate interior angles are on opposite sides of the transversal, inside the parallel lines — they’re equal.
- Vertical angles are opposite each other when two lines cross — always equal.
- Supplementary angles add up to 180° — like adjacent angles on a straight line.
- Complementary angles add up to 90° — not common here unless specified.

Looking at the diagram:

Line X is top horizontal line, Line Y is bottom horizontal line. Transversal Z crosses them diagonally from top-left to bottom-right.

Angles around intersection with line X:
Top-left = ∠1, Top-right = ∠2, Bottom-left = ∠3, Bottom-right = ∠4

Angles around intersection with line Y:
Top-left = ∠5, Top-right = ∠6, Bottom-left = ∠7, Bottom-right = ∠8

So positions:
∠1 and ∠5 → both top-left → corresponding
∠2 and ∠6 → both top-right → corresponding
∠3 and ∠7 → both bottom-left → corresponding
∠4 and ∠8 → both bottom-right → corresponding

Also:
∠3 and ∠6 → alternate interior (inside, opposite sides) → equal
∠4 and 5 → alternate interior → equal
∠1 and ∠4 → vertical → equal
∠2 and ∠3 → vertical → equal
∠5 and ∠8 → vertical → equal
∠6 and ∠7 → vertical → equal

Adjacent angles on a straight line (like ∠1 + ∠2, or ∠3 + ∠4) = 180°

Now let’s answer each question one by one.

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1. Angles 4 and 5 are which type of angle pair?

∠4 is bottom-right at top line.
∠5 is top-left at bottom line.
They are on opposite sides of the transversal, and between the two parallel lines → that’s alternate interior angles.

✔ Answer: Alternate interior angles

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2. True or false: Angle 1 and Angle 5 are corresponding angles.

∠1 is top-left at top line.
∠5 is top-left at bottom line.
Same relative position → yes, corresponding.

✔ Answer: True

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3. Angles 2 and 3 are which type of angle pair?

∠2 and 3 are next to each other at the same intersection (top line), forming a straight line → they are adjacent angles on a straight line, so supplementary. But also, they are vertical? No — wait, actually, looking again:

At the top intersection:
∠1 and 2 are adjacent (form straight line)
∠2 and 4 are adjacent? Wait no — better to think:

Actually, ∠2 and ∠3 are not vertical. Vertical would be ∠1 & ∠4, ∠2 & 3? Let me check:

When two lines cross, vertical angles are opposite. So at top intersection:

Lines X and Z cross → four angles: ∠1, 2, ∠3, 4.

If ∠1 is top-left, then going clockwise: ∠2 (top-right), ∠4 (bottom-right), ∠3 (bottom-left).

So vertical pairs: ∠1 & ∠4, ∠2 & ∠3.

Yes! So ∠2 and ∠3 are vertical angles.

Wait — but in standard labeling, if you have two lines crossing, vertical angles are opposite. So if ∠2 is top-right, its vertical partner should be bottom-left, which is ∠3. Yes.

But sometimes people get confused — let me confirm with values later. For now, based on position: ∠2 and 3 are vertical angles.

BUT — hold on! In many diagrams, if ∠1, ∠2, ∠3, ∠4 are labeled around the point, often ∠1 and ∠3 are vertical, ∠2 and 4 are vertical. I need to be careful.

Looking back at typical setup:

Usually, for two lines intersecting:

Label as:
∠1 ∠2
\ /
X
/ \
∠3 4

Then vertical: ∠1 & ∠4, ∠2 & ∠3? Or ∠1 & ∠3?

Standard: If lines cross, vertical angles are directly opposite.

In most textbooks, if you label consecutively around the point, ∠1 and 3 are vertical, ∠2 and ∠4 are vertical.

I think I made a mistake.

Let me redefine clearly.

Assume at the top intersection (line X and transversal Z):

Imagine standing at the intersection. The four angles formed:

- Above line X, left of Z: ∠1
- Above line X, right of Z: ∠2
- Below line X, left of Z: ∠3
- Below line X, right of Z: ∠4

Then:

∠1 and 4 are NOT vertical — because ∠1 is above-left, ∠4 is below-right — those ARE vertical! Because they are opposite.

Similarly, ∠2 (above-right) and ∠3 (below-left) are vertical.

Yes — so ∠2 and ∠3 are vertical angles.

But in some systems, it's labeled differently. However, given the context and standard problems, we'll go with:

∠2 and 3 are vertical angles.

But let’s verify with logic: if ∠2 and ∠3 were adjacent, they’d sum to 180°, but vertical angles are equal.

In this case, since they are opposite each other at the intersection, they are vertical.

✔ Answer: Vertical angles

Wait — I recall now: in many diagrams for parallel lines, angles are labeled such that:

At top line: ∠1, 2 on top side; ∠3, ∠4 on bottom side.

And typically, ∠1 and ∠3 are on the same side of transversal? No.

Perhaps I should look at the actual diagram description.

The user didn’t provide coordinates, but from standard problems:

Usually, for transversal cutting two parallels:

Top line intersections: angles 1,2,3,4 where 1 and 2 are on one side, etc.

To avoid confusion, let’s use the fact that in such diagrams, angles 2 and 3 are often adjacent and form a linear pair.

Let me think differently.

From the questions later, we can infer.

For example, question: "True or false: Angle 2 and Angle 3 are supplementary" — but that’s not asked yet.

Question 3 is: "Angles 2 and 3 are which type of angle pair?"

In standard geometry, when two lines intersect, the adjacent angles are supplementary, and vertical angles are equal.

At the intersection of line X and transversal Z, angles 2 and 3 are likely adjacent if labeled sequentially.

Assume the labeling is:

As you go around the point, starting from top-left: ∠1, then moving clockwise: ∠2 (top-right), then ∠4 (bottom-right), then ∠3 (bottom-left). That doesn't make sense.

Better to assume:

Typical labeling for such diagrams:

- At top intersection:
- ∠1: upper left
- ∠2: upper right
- ∠3: lower left
- ∠4: lower right

Then:
- ∠1 and ∠4 are vertical? No — ∠1 (upper left) and ∠4 (lower right) are vertical.
- ∠2 (upper right) and ∠3 (lower left) are vertical.

Yes.

And ∠1 and ∠2 are adjacent (sum 180°), ∠2 and ∠4 are adjacent, etc.

So ∠2 and 3 are vertical angles.

But let's confirm with a different approach.

Later question: "If m∠3 = 123°, find m∠8"

If ∠3 and 8 are related.

∠3 is at top line, lower left.
∠8 is at bottom line, lower right.

Not obviously related.

Perhaps I should proceed and see consistency.

Another way: in many textbooks, for parallel lines cut by transversal, the angles are labeled as:

Line X (top): angles 1,2 on the top side of X, but usually it's per intersection.

I found a better way: let's list all angle pairs based on standard knowledge.

From memory, in such a diagram:

- Corresponding: (1,5), (2,6), (3,7), (4,8)
- Alternate interior: (3,6), (4,5)
- Alternate exterior: (1,8), (2,7)
- Vertical: (1,4), (2,3), (5,8), (6,7) — wait, is that right?

If at each intersection, vertical angles are opposite.

At top intersection: if ∠1 and ∠4 are opposite, then yes.

But if ∠1 is top-left, ∠4 is bottom-right, they are opposite, so vertical.

Similarly, ∠2 (top-right) and ∠3 (bottom-left) are vertical.

Yes.

So for question 3: Angles 2 and 3 are vertical angles.

But let's double-check with the next question.

Question 4: Name all angles congruent to angle 6.

∠6 is at bottom line, top-right.

Congruent angles to ∠6:

- Corresponding: ∠2 (since 2 and 6 are corresponding)
- Vertical: ∠7 (at bottom line, bottom-left? No)

At bottom intersection:
∠5: top-left
∠6: top-right
∠7: bottom-left
∠8: bottom-right

So vertical to ∠6 is ∠7? No — ∠6 (top-right) and ∠7 (bottom-left) are not vertical; vertical would be ∠6 and the angle opposite, which is ∠7 only if labeled that way.

Standard: at a point, vertical angles are across from each other.

So if ∠6 is top-right, its vertical partner is bottom-left, which is ∠7.

Is that correct? Let's think: when two lines cross, the vertical angles are the ones not sharing a side.

So for bottom intersection: lines Y and Z cross.

Angles:
- Between Y-top and Z-left: ∠5
- Between Y-top and Z-right: ∠6
- Between Y-bottom and Z-left: ∠7
- Between Y-bottom and Z-right: ∠8

Then vertical pairs: ∠5 and ∠8 (opposite), ∠6 and ∠7 (opposite).

Yes! So ∠6 and ∠7 are vertical angles, so congruent.

Also, corresponding to ∠6 is ∠2 (both top-right).

Also, alternate interior to ∠6 is 3 (because ∠3 is bottom-left at top line, ∠6 is top-right at bottom line — wait, alternate interior are on opposite sides of transversal and inside the parallels.

Inside means between the two parallel lines.

So for ∠6 (at bottom line, top-right), it is on the "inside" if we consider the region between X and Y.

Transversal Z, so angles between X and Y are interior.

So ∠3, ∠4, ∠5, ∠6 are interior angles.

Alternate interior: on opposite sides of transversal.

So ∠3 (left side, bottom of top line) and ∠6 (right side, top of bottom line) — are they alternate interior? Let's see:

∠3 is on the left side of Z, below X.
∠6 is on the right side of Z, above Y.

Since X and Y are parallel, and Z is transversal, the alternate interior angles are:

- ∠3 and ∠6: ∠3 is interior, left side; ∠6 is interior, right side — yes, alternate interior, so equal.

Similarly, ∠4 and ∠5 are alternate interior.

So for ∠6, congruent angles are:

- Corresponding: ∠2
- Vertical: ∠7
- Alternate interior: ∠3

Is that all? Also, since ∠2 = ∠6 (corresponding), and ∠2 has vertical ∠3, so ∠3 = ∠2 = ∠6, and ∠7 = ∠6 (vertical), so yes.

So angles congruent to ∠6: ∠2, ∠3, ∠7

Because:
- ∠2 ≅ ∠6 (corresponding)
- ∠3 ≅ ∠2 (vertical at top) so ∠3 ≅ ∠6
- ∠7 ≅ ∠6 (vertical at bottom)

Also, is there more? ∠4 is not necessarily equal, etc.

So for question 4: Name all angles congruent to angle 6.

Answer: ∠2, 3, ∠7

Now back to question 3: Angles 2 and 3.

At top intersection, ∠2 and ∠3 are vertical angles, as established.

So answer is vertical angles.

But let's confirm with calculation later.

Proceed.

4. Name all the angles that are congruent to angle 6.

As above: ∠2 (corresponding), ∠3 (since ∠3 ≅ ∠2 via vertical, and ∠2 ≅ ∠6), and ∠7 (vertical to ∠6).

Also, is ∠4 congruent? ∠4 is alternate interior to ∠5, not directly to ∠6.

∠4 and ∠6 are not necessarily equal.

For example, if ∠6 is 60°, then ∠2=60°, ∠3=60° (vertical to ∠2), ∠7=60° (vertical to ∠6), while ∠4 might be 120° if adjacent.

So only ∠2, ∠3, ∠7.

But ∠3 is congruent to ∠6 because both are equal to ∠2.

Yes.

So answer: angles 2, 3, and 7.

5. Give two examples of corresponding angles.

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

Examples:
- ∠1 and ∠5 (both top-left)
- ∠2 and 6 (both top-right)
- ∠3 and ∠7 (both bottom-left)
- ∠4 and 8 (both bottom-right)

Any two of these.

Say: ∠1 and ∠5, or ∠2 and ∠6.

6. True or false: Angle 2 and Angle 3 are supplementary angles.

Supplementary means add to 180°.

At the top intersection, ∠2 and 3 are vertical angles, so they are equal, not necessarily supplementary.

Unless they are 90° each, but generally not.

Are they adjacent? In our labeling, ∠2 and ∠3 are not adjacent; they are vertical.

Adjacent to ∠2 would be ∠1 and ∠4.

For example, ∠2 and ∠1 are adjacent and supplementary.

∠2 and ∠4 are adjacent and supplementary.

But ∠2 and ∠3 are vertical, so if the lines are not perpendicular, they are not 90°, so not supplementary.

In general, vertical angles are equal, and only supplementary if each is 90°.

But the statement is "are supplementary", implying always, which is false.

Moreover, in the context, since no specific measure, we assume general case.

So false.

But let's see: if ∠2 and ∠3 are vertical, and say ∠2 = x, then ∠3 = x, and x + x = 2x = 180° only if x=90°, which is not given.

So generally false.

However, I think I have a labeling issue.

Perhaps in the diagram, angles 2 and 3 are adjacent.

Let me search for a standard.

Upon second thought, in many worksheets, for the top intersection, angles are labeled as:

- ∠1 and ∠2 on the top side of line X, but that doesn't make sense.

Perhaps the labeling is sequential around the point.

Assume that at the top intersection, the angles are labeled in order: ∠1, 2, ∠3, 4 going clockwise.

So if ∠1 is top-left, then ∠2 is top-right, ∠3 is bottom-right, ∠4 is bottom-left.

Then:
- Vertical: ∠1 and ∠3, ∠2 and ∠4
- Adjacent: ∠1 and ∠2, ∠2 and ∠3, etc.

In this case, ∠2 and ∠3 are adjacent and form a straight line, so supplementary.

This is more common in some texts.

For example, if you have a cross, and label 1,2,3,4 clockwise, then 1 and 3 are vertical, 2 and 4 are vertical, and 1+2=180, 2+3=180, etc.

So perhaps in this problem, that's the case.

Let me check with the first question.

Question 1: Angles 4 and 5 are which type?

If at top, ∠4 is bottom-left (if labeled clockwise from top-left: 1=top-left, 2=top-right, 3=bottom-right, 4=bottom-left)

Then ∠4 is bottom-left at top line.

∠5 is at bottom line, top-left.

So ∠4 and ∠5: ∠4 is below X, left of Z; ∠5 is above Y, left of Z.

So they are on the same side of the transversal (left), and both are "interior" if we consider between the lines, but ∠4 is below X, which is inside if Y is below, so yes, both are on the left side, and between the parallels? ∠4 is between X and Y? Since X is top, Y is bottom, and Z is diagonal, the region between X and Y is the strip.

∠4 is at top line, below it, so in the strip. ∠5 is at bottom line, above it, so in the strip. And both on the left side of Z.

So they are on the same side of the transversal, and both interior — that would be consecutive interior or same-side interior, which are supplementary.

But the question asks for "which type", and in many contexts, they are called "same-side interior angles".

But earlier I said alternate interior, but that's for opposite sides.

For ∠4 and 5: if both on left side, then same-side interior.

But in standard, alternate interior are on opposite sides.

So if ∠4 is left side, ∠5 is left side, then not alternate.

Let's define:

- Interior angles: between the two parallel lines: so for top line, the angles below it (∠3 and ∠4 if labeled as above); for bottom line, the angles above it (∠5 and ∠6).

So interior angles are ∠3, ∠4, ∠5, ∠6.

Now, for transversal Z, the left side and right side.

Assume Z is coming from top-left to bottom-right, so left side is the side where the acute angle might be, but let's say the side towards the left of the page.

So for a point on Z, the left side is one half-plane.

Typically, for angle at top line, ∠3 and ∠4: if ∠3 is on the left, ∠4 on the right, or vice versa.

To resolve, let's look at the corresponding angles mentioned in question 2.

Question 2: True or false: Angle 1 and Angle 5 are corresponding angles.

If ∠1 is at top-left, ∠5 at bottom-left, then yes, corresponding.

So likely, ∠1 and ∠5 are both on the left side, top and bottom.

So for the top intersection, ∠1 is top-left, so probably the labeling is:

- ∠1: top-left
- ∠2: top-right
- ∠3: bottom-left
- ∠4: bottom-right

Then at bottom intersection:
- ∠5: top-left
- ∠6: top-right
- ∠7: bottom-left
- ∠8: bottom-right

Then:
- Corresponding: (1,5), (2,6), (3,7), (4,8)
- Alternate interior: (3,6) and (4,5) [because 3 is bottom-left at top, 6 is top-right at bottom — opposite sides of transversal]
- Same-side interior: (3,5) and (4,6) [same side]

For question 1: Angles 4 and 5.

∠4 is bottom-right at top line.
∠5 is top-left at bottom line.

So ∠4 is on the right side of Z, below X.
∠5 is on the left side of Z, above Y.

So they are on opposite sides of the transversal, and both are interior (between the lines).

So they are alternate interior angles.

Yes.

And for question 3: Angles 2 and 3.

∠2 is top-right at top line.
∠3 is bottom-left at top line.

At the same intersection, so they are not both at different lines.

At the top intersection, ∠2 and 3 are not adjacent; they are on opposite corners.

Specifically, ∠2 is top-right, ∠3 is bottom-left, so they are vertical angles.

Because when two lines cross, the vertical angles are the ones not sharing a ray.

So ∠2 and 3 are vertical angles.

And in this case, they are equal.

For question 6: True or false: Angle 2 and Angle 3 are supplementary.

Since they are vertical, they are equal, and their sum is 2*measure, which is 180° only if each is 90°, which is not specified, so generally false.

But in the context of the diagram, if no right angle, false.

Moreover, in the options, it's a true/false with boxes, so likely false.

But let's see the calculation questions to verify.

Question: If m∠3 = 123°, find m∠8.

∠3 = 123°.

∠3 is at top line, bottom-left.

∠8 is at bottom line, bottom-right.

How are they related?

First, ∠3 and ∠7 are corresponding? ∠3 is bottom-left at top, ∠7 is bottom-left at bottom, so yes, corresponding angles, so ∠7 = ∠3 = 123°.

Then ∠7 and ∠8 are adjacent on the straight line at bottom intersection, so ∠7 + ∠8 = 180°.

So 123° + m∠8 = 180°, so m∠8 = 57°.

If we had assumed something else, it might not work.

Another way: ∠3 and 6 are alternate interior, so 6 = ∠3 = 123°.

Then ∠6 and ∠8 are vertical? At bottom, ∠6 is top-right, ∠8 is bottom-right, so not vertical; vertical to ∠6 is ∠7.

∠6 and 8 are adjacent if on the same side, but in our labeling, at bottom, ∠6 and ∠8 are not adjacent; ∠6 and ∠7 are adjacent, ∠7 and 8 are adjacent, etc.

From above, with ∠3 = 123°, ∠7 = 123° (corresponding), then ∠8 = 180° - 123° = 57° since ∠7 and 8 are on a straight line.

Yes.

Now for angles 2 and 3: if ∠3 = 123°, and if ∠2 and ∠3 are vertical, then ∠2 = 123°, but then at the top intersection, ∠2 and ∠1 are adjacent, so ∠1 = 180° - 123° = 57°, and so on.

But are ∠2 and 3 supplementary? 123° + 123° = 246° ≠ 180°, so not supplementary.

So for question 6, it is false.

But in some labelings, if ∠2 and ∠3 are adjacent, then they would be supplementary.

Given that with this labeling, the calculations work, and for question 1, angles 4 and 5 are alternate interior, which matches, I'll stick with this labeling.

So summary:

- ∠1: top-left top
- ∠2: top-right top
- ∠3: bottom-left top
- ∠4: bottom-right top
- ∠5: top-left bottom
- ∠6: top-right bottom
- ∠7: bottom-left bottom
- ∠8: bottom-right bottom

Vertical angles:
- At top: ∠1 & ∠4, ∠2 & ∠3
- At bottom: ∠5 & ∠8, ∠6 & ∠7

Corresponding: (1,5), (2,6), (3,7), (4,8)

Alternate interior: (3,6), (4,5) [since 3 and 6 are on opposite sides of transversal and interior]

Same-side interior: (3,5), (4,6)

Now answer each question.

1. Angles 4 and 5 are which type of angle pair?

∠4: bottom-right top
∠5: top-left bottom

So ∠4 is on the right side of Z, below X.
∠5 is on the left side of Z, above Y.

Opposite sides of transversal, and both interior (between the parallels), so alternate interior angles.

✔ Answer: Alternate interior angles

2. True or false: Angle 1 and Angle 5 are corresponding angles.

∠1: top-left top
∠5: top-left bottom
Same relative position, so yes.

✔ Answer: True

3. Angles 2 and 3 are which type of angle pair?

∠2: top-right top
∠3: bottom-left top
At the same intersection, and they are vertical angles (opposite each other).

✔ Answer: Vertical angles

4. Name all the angles that are congruent to angle 6.

∠6: top-right bottom

Congruent to:
- Corresponding: ∠2 (top-right top)
- Vertical: ∠7 (bottom-left bottom) — because at bottom, ∠6 and 7 are vertical? No.

At bottom intersection:
∠5: top-left
∠6: top-right
∠7: bottom-left
∠8: bottom-right

Vertical pairs: ∠5 and ∠8 (opposite), ∠6 and ∠7 (opposite)? Is ∠6 opposite to ∠7?

If ∠6 is top-right, its opposite should be bottom-left, which is ∠7, yes.

So ∠6 ∠7 (vertical)

Also, corresponding to ∠6 is ∠2, so ∠2 ≅ ∠6

Also, since ∠2 ≅ ∠3 (vertical at top), so ∠3 ≅ ∠6

Is there more? ∠4 is not necessarily equal.

So angles congruent to ∠6: 2, ∠3, 7

Because:
- 2 ≅ ∠6 (corresponding)
- ∠3 ≅ ∠2 (vertical) so ∠3 ≅ ∠6
- ∠7 ≅ ∠6 (vertical)

Yes.

✔ Answer: angles 2, 3, and 7

5. Give two examples of corresponding angles.

Any two pairs: e.g., ∠1 and 5, or ∠2 and ∠6, or ∠3 and ∠7, or ∠4 and ∠8.

Say: ∠1 and ∠5, and ∠2 and ∠6.

✔ Answer: ∠1 and 5; ∠2 and 6 (or any two corresponding pairs)

6. True or false: Angle 2 and Angle 3 are supplementary angles.

As established, they are vertical angles, so equal, and their sum is not necessarily 180°. In fact, from earlier, if ∠3=123°, ∠2=123°, sum=246°≠180°, so false.

Even without numbers, vertical angles are not supplementary unless right angles.

So false.

✔ Answer: False

Now the calculation questions.

7. If m∠3 = 123°, find m∠8.

∠3 = 123°

∠3 and ∠7 are corresponding angles (both bottom-left), so ∠7 = ∠3 = 123°

At the bottom intersection, ∠7 and ∠8 are adjacent angles on a straight line (since they form a linear pair along line Y), so ∠7 + 8 = 180°

Thus, 123° + m∠8 = 180°

m∠8 = 180° - 123° = 57°

✔ Answer: 57°

8. If m∠4 = 47°, find m∠6.

∠4 = 47°

∠4 and ∠6: what relationship?

∠4 is bottom-right top
∠6 is top-right bottom

They are on the same side of the transversal (right side), and both are interior? ∠4 is below X, so interior; ∠6 is above Y, so interior. Same side, so same-side interior angles, which are supplementary.

Same-side interior angles are supplementary when lines are parallel.

So ∠4 + ∠6 = 180°

Thus, 47° + m∠6 = 180°

m∠6 = 180° - 47° = 133°

Alternatively, ∠4 and ∠5 are alternate interior? ∠4 is right side, ∠5 is left side, so not.

∠4 and ∠8 are corresponding, so ∠8 = 4 = 47°

Then ∠8 and ∠6 are vertical? At bottom, ∠8 is bottom-right, ∠6 is top-right, so not vertical; vertical to ∠8 is ∠5.

∠6 and 8 are adjacent? In our labeling, at bottom, ∠6 and ∠8 are not directly adjacent; ∠6 and 7 are adjacent, ∠7 and ∠8 are adjacent.

∠6 and 8 are on the same side, but not adjacent.

From corresponding: ∠4 ≅ ∠8 (corresponding), so ∠8 = 47°

Then ∠8 and ∠6: are they related? ∠6 and ∠8 are both at bottom, but ∠6 is top-right, ∠8 is bottom-right, so they are adjacent if we consider the line, but actually, at the intersection, ∠6 and 8 share the ray along Y, but are on opposite sides of Z.

Actually, ∠6 and 8 are adjacent angles that form a straight line with the transversal? No.

The angles around the point sum to 360°, but specifically, ∠6 and ∠8 are not necessarily supplementary.

Note that ∠6 and ∠7 are vertical? Earlier we said ∠6 and ∠7 are vertical, but let's confirm.

At bottom intersection:
- ∠5 and ∠8 are vertical (top-left and bottom-right)
- ∠6 and 7 are vertical (top-right and bottom-left)

Yes.

So ∠6 ≅ ∠7

And ∠7 and ∠8 are adjacent on line Y, so ∠7 + ∠8 = 180°

Since ∠8 = 4 = 47° (corresponding), then ∠7 = 180° - 47° = 133°

Then ∠6 = ∠7 = 133°

Same as before.

From same-side interior: ∠4 and ∠6 are same-side interior? ∠4 is at top, right side; ∠6 is at bottom, right side; so same side of transversal, and both interior, so yes, same-side interior angles, supplementary.

So ∠4 + ∠6 = 180°, so 47° + ∠6 = 180°, ∠6 = 133°

✔ Answer: 133°

9. If m∠1 = 161°, find m∠5.

∠1 and ∠5 are corresponding angles, so ∠5 = ∠1 = 161°

Is that it? But let's see if it makes sense.

∠1 = 161°, then at top, ∠1 and ∠2 are adjacent, so ∠2 = 180° - 161° = 19°

Then ∠5 = ∠1 = 161° (corresponding)

At bottom, ∠5 and ∠6 are adjacent, so ∠6 = 180° - 161° = 19°, which matches ∠2 = 19°, good.

So yes.

✔ Answer: 161°

10. If m∠7 = 112°, find m∠8.

∠7 and ∠8 are adjacent angles on the straight line at bottom intersection, so they are supplementary.

∠7 + 8 = 180°

112° + m∠8 = 180°

m∠8 = 180° - 112° = 68°

Also, ∠7 and 3 are corresponding, so ∠3 = 112°, etc., but not needed.

✔ Answer: 68°

11. If m∠5 = 71°, find m∠4.

∠5 = 71°

∠5 and ∠4: what relationship?

∠5 is top-left bottom
∠4 is bottom-right top

They are alternate interior angles? ∠5 is left side, ∠4 is right side, and both interior, so yes, alternate interior angles, so congruent.

Thus, ∠4 = ∠5 = 71°

Confirm: alternate interior angles are equal when lines are parallel.

Yes.

∠5 and ∠1 are corresponding, so ∠1 = 71°

Then ∠1 and ∠4 are vertical? At top, ∠1 and ∠4 are vertical, so ∠4 = ∠1 = 71°, same thing.

So yes.

✔ Answer: 71°

Now, to box the answers.

But the user has multiple questions, so I need to provide all answers.

Also, for the true/false, there are checkboxes, but since it's text, I'll write true or false.

For the "name all angles", list them.

So final answers:

1. Alternate interior angles
2. True
3. Vertical angles
4. Angles 2, 3, and 7
5. For example, ∠1 and 5; ∠2 and ∠6 (any two corresponding pairs)
6. False
7. 57°
8. 133°
9. 161°
10. 68°
11. 71°

For question 5, since it says "give two examples", I can specify two pairs.

In the response, I'll list them clearly.

Now for the Final Answer section, since there are multiple parts, I need to present all.

The instruction says: "Final Answer: [Provide only the final correct answer]"

But since there are 11 questions, I should list them.

Perhaps number them.

Since it's a worksheet, I'll provide the answers in order.

So:

Final Answer:
1. Alternate interior angles
2. True
3. Vertical angles
4. 2, 3, 7
5. ∠1 and 5; ∠2 and 6 (or other corresponding pairs)
6. False
7. 57
8. 133
9. 161
10. 68
11. 71

For question 4, "name all the angles", so list the numbers.

For question 5, since it's examples, I can say "for instance, angle 1 and angle 5, and angle 2 and angle 6"

But to be concise.

In the boxed answer, perhaps just the values.

Since it's text, I'll write it out.

I think it's fine.

Note: for question 6, it's "True or false", and in the image, there are boxes for True and False, but in answer, we say False.

Similarly for others.

So I'll proceed.