Find angles and variables in parallel line problems with transversals.

Let’s solve each problem step by step.

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Problem 1:

We have two parallel lines cut by a transversal. The angle marked 60° and ∠1 are vertical angles, so they are equal → m∠1 = 60°.

∠1 and ∠2 are same-side interior angles (on the same side of the transversal, between the parallel lines). These add up to 180°.

So:
m∠2 = 180° - 60° = 120°

✔ Final for #1: m∠1 = 60°, m∠2 = 120°

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Problem 2:

The 40° angle and ∠2 are vertical angles → m∠2 = 40°.

∠1 and ∠2 are corresponding angles? Wait — actually, look: ∠1 is on top line, ∠2 is on bottom line, both on the same side of the transversal → they are corresponding angles if lines are parallel. But wait — in this diagram, the arrows show the lines are parallel.

Actually, ∠1 and the 40° angle are alternate exterior angles? Let me think again.

Better approach: The 40° angle and ∠2 are vertical → ∠2 = 40°.

Now, ∠1 and ∠2 are supplementary? No — they’re on opposite sides.

Wait — actually, ∠1 and the 40° angle are corresponding angles because they’re in matching positions relative to the transversal and parallel lines → so ∠1 = 40°? That can’t be right because they’re not in corresponding positions.

Let me redraw mentally:

Top line: angle labeled “1” is above the line, on the right side of transversal.

Bottom line: angle labeled “2” is below the line, on the left side of transversal? Wait no — looking at standard position:

Actually, in problem 2, the 40° angle is adjacent to ∠2 — they form a linear pair? No — they are vertical? Actually, yes — ∠2 and 40° are vertical angles → so ∠2 = 40°.

Then ∠1 and ∠2 are alternate interior angles? If lines are parallel, alternate interior angles are equal → so ∠1 = ∠2 = 40°? But that would mean ∠1 = 40°, but visually it looks obtuse.

Wait — I think I misread. Let me check again.

In problem 2: The transversal crosses two parallel lines. On the bottom line, there's an angle marked 40°, and next to it is ∠2 — they are adjacent and form a straight line? Or are they vertical?

Looking at typical diagrams: usually, when two lines cross, vertical angles are opposite. So if 40° is one angle, then the angle directly opposite is also 40° — that’s ∠2? Then ∠2 = 40°.

Then ∠1 is on the top line, same side as the 40° angle — so ∠1 and the 40° angle are corresponding angles → so ∠1 = 40°? But that doesn't make sense with the drawing.

Wait — perhaps ∠1 and ∠2 are supplementary? Let’s use logic.

Actually, let’s label:

Assume the transversal goes from bottom-left to top-right.

On the bottom line: the angle between the transversal and the line on the right side is 40°. Then ∠2 is the angle on the left side of the transversal on the bottom line — so ∠2 and 40° are adjacent and form a straight line → so ∠2 = 180° - 40° = 140°.

Then ∠1 is on the top line, on the left side — so ∠1 and ∠2 are corresponding angles → since lines are parallel, ∠1 = ∠2 = 140°.

Yes! That makes sense.

So:
m∠2 = 180° - 40° = 140° (linear pair)
m∠1 = m∠2 = 140° (corresponding angles)

✔ Final for #2: m∠1 = 140°, m∠2 = 140°

Wait — that can’t be right because if ∠1 and ∠2 are both 140°, and they’re on different lines, but corresponding angles should be equal — yes, that’s fine.

But let me confirm: if the 40° is on the bottom right, then the angle next to it on the bottom left is 140° — that’s ∠2. Then on the top line, the angle on the top left (∠1) corresponds to ∠2 — so yes, ∠1 = 140°.

Correct.

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Problem 3:

Angle marked 45° and ∠2 are vertical angles → m∠2 = 45°.

∠1 and ∠2 are adjacent angles forming a straight line → so m∠1 + m∠2 = 180°

Thus: m∠1 = 180° - 45° = 135°

✔ Final for #3: m∠1 = 135°, m∠2 = 45°

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Problem 4:

Angle marked 75° and ∠2 are adjacent on a straight line → so m∠2 = 180° - 75° = 105°

∠1 and ∠2 are corresponding angles (since lines are parallel) → so m∠1 = m∠2 = 105°

Wait — are they corresponding? Let’s see: ∠2 is on top line, right side; ∠1 is on bottom line, right side — yes, corresponding → equal.

✔ Final for #4: m∠1 = 105°, m∠2 = 105°

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Problem 5:

Two intersecting lines. One angle is 80°, another is x°, and y° is adjacent.

First, x° and 80° are vertical angles → so x = 80

Then, x° and y° are adjacent angles forming a straight line → so x + y = 180

80 + y = 180 → y = 100

✔ Final for #5: x = 80, y = 100

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Problem 6:

Perpendicular lines — the square means 90°.

x° and y° are adjacent angles forming a right angle? Wait — the vertical line is perpendicular to horizontal lines.

Actually, the figure shows two horizontal parallel lines cut by a vertical transversal, which is perpendicular → so all angles are 90°.

But x° and y° are shown — x is on top, y is on bottom, both on the same side.

Since the transversal is perpendicular, x = 90°, y = 90°

Also, x and y are corresponding angles → equal, and since perpendicular, both 90°.

✔ Final for #6: x = 90, y = 90

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Problem 7:

Two vertical parallel lines cut by a transversal.

Angles: 80° and 2x° — these are same-side interior angles → sum to 180°

So: 80 + 2x = 180

2x = 100

x = 50

✔ Final for #7: x = 50

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Problem 8:

Two parallel lines cut by a transversal.

Angles: 2x° and 110° — these are same-side interior angles → sum to 180°

So: 2x + 110 = 180

2x = 70

x = 35

✔ Final for #8: x = 35

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Problem 9:

Parallel lines cut by transversal.

Angles: 100° and (x - 10)° — these are corresponding angles? Let’s see position.

If 100° is on top left, and (x-10) is on bottom left — then they are corresponding → equal.

So: x - 10 = 100

x = 110

✔ Final for #9: x = 110

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Problem 10:

Parallel lines cut by transversal.

Angles: 60° and (3x + 15)° — these are same-side interior angles? Or alternate?

Looking: 60° is on top, left side; (3x+15) is on bottom, right side — so they are alternate interior angles → equal.

Is that correct? Alternate interior are on opposite sides of transversal, inside the parallel lines.

Yes — so 3x + 15 = 60

3x = 45

x = 15

✔ Final for #10: x = 15

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Now, compiling all answers:

1. m∠1 = 60°, m∠2 = 120°
2. m∠1 = 140°, m∠2 = 140°
3. m∠1 = 135°, m∠2 = 45°
4. m∠1 = 105°, m∠2 = 105°
5. x = 80, y = 100
6. x = 90, y = 90
7. x = 50
8. x = 35
9. x = 110
10. x = 15

Final Answer:
1. m∠1 = 60°, m∠2 = 120°
2. m∠1 = 140°, m∠2 = 140°
3. m∠1 = 135°, m∠2 = 45°
4. m∠1 = 105°, m∠2 = 105°
5. x = 80, y = 100
6. x = 90, y = 90
7. x = 50
8. x = 35
9. x = 110
10. x = 15