Find the value of x in each figure where two parallel lines are cut by a transversal, focusing on corresponding angles.

Let's solve each of these problems step by step using the properties of corresponding angles, supplementary angles, and parallel lines cut by a transversal.

When two parallel lines are cut by a transversal, several angle relationships apply:

- Corresponding angles are equal.
- Alternate interior angles are equal.
- Same-side interior angles are supplementary (add to 180°).
- Vertical angles are equal.
- Linear pairs add up to 180°.

We'll go through each figure one at a time.

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1)

Given: Two parallel lines, transversal. Angles: 146°, 34°, and $ x $ is opposite to 34°.

Looking at the diagram:
- The angle marked 34° and $ x $ are vertical angles → so they are equal.
- Also, 146° + 34° = 180°, so they form a linear pair.

So $ x = 34^\circ $

✔ Answer: $ x = 34^\circ $

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2)

Angles: 93° and 87° on the same side of the transversal between two parallel lines.

- These two angles are on the same side of the transversal and between the lines → same-side interior angles.
- They should be supplementary if the lines are parallel.

But 93° + 87° = 180° → yes, they are supplementary.

Now, $ x $ is adjacent to 93°, forming a linear pair with it? Wait — actually, look closely:

- The 93° and 87° are on the same side of the transversal, but not necessarily adjacent.
- But $ x $ is on the other side of the transversal, and seems to be corresponding or alternate to 87°?

Wait — let’s analyze:

The 93° and 87° are adjacent angles along the transversal — their sum is 180°, so they form a straight line.

Then $ x $ is vertically opposite to 93° → so $ x = 93^\circ $

Alternatively, $ x $ and 93° are vertical angles → so $ x = 93^\circ $

✔ Answer: $ x = 93^\circ $

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3)

Two parallel lines, transversal. One angle is 75°, another is 105°, and $ x $ is opposite to 75°.

Wait — 75° and 105° are on the same side of the transversal?

Actually, the diagram shows:
- A transversal cutting two parallel lines.
- At the top line: 75° and 105° are adjacent angles → 75° + 105° = 180° → they form a linear pair.
- So they are on a straight line.

Now, $ x $ is on the bottom line, and it's corresponding to the 75° angle? Or alternate?

Wait — the 75° and $ x $ are in matching positions relative to the transversal and the lines → so they are corresponding angles.

Since lines are parallel, corresponding angles are equal → $ x = 75^\circ $

✔ Answer: $ x = 75^\circ $

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4)

Parallel lines, transversal. One angle is 111°, and $ x $ is adjacent to it.

- 111° and $ x $ are on a straight line → linear pair
- So $ x + 111^\circ = 180^\circ $
- $ x = 180 - 111 = 69^\circ $

Also, $ x $ is corresponding to the angle opposite 111° — but we don’t need that.

✔ Answer: $ x = 69^\circ $

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5)

One angle is 150°, another is 30°, and $ x $ is on the upper line.

Note: 150° and 30° are on the same side of the transversal.

Wait — 150° + 30° = 180° → so they are supplementary.

But more importantly: $ x $ is corresponding to the 30° angle? Let’s see.

The 30° angle is on the lower line, and $ x $ is on the upper line, in the same relative position → so they are corresponding angles.

So $ x = 30^\circ $

✔ Answer: $ x = 30^\circ $

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6)

Two parallel lines, transversal. One angle is 52°, and $ x $ is opposite to it.

- The 52° and $ x $ are vertical angles → so $ x = 52^\circ $

Alternatively, $ x $ and 52° are vertically opposite → equal.

✔ Answer: $ x = 52^\circ $

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7)

One angle is 131°, another is 49°, and $ x $ is on the upper line.

Note: 131° + 49° = 180° → so they are supplementary.

Now, $ x $ is corresponding to the 49° angle? Or alternate?

Look: 49° is on the lower line, $ x $ is on the upper line, and both are on the same side of the transversal → so they are corresponding angles.

So $ x = 49^\circ $

Alternatively, 131° and $ x $ are same-side interior angles → they should add to 180°

So $ x = 180 - 131 = 49^\circ $

✔ Answer: $ x = 49^\circ $

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8)

One angle is 103°, another is 77°, and $ x $ is on the left side.

Note: 103° + 77° = 180° → so they are supplementary.

Now, $ x $ is vertical to 77°? Or adjacent?

Wait — $ x $ is on the left side of the transversal, and 103° is on the right.

But $ x $ and 103° are alternate interior angles? Or corresponding?

Wait — $ x $ and 77° are vertical angles? No.

Wait — 77° and $ x $ are on opposite sides of the transversal and between the lines → alternate interior angles.

But since the lines are parallel, alternate interior angles are equal.

So $ x = 77^\circ $

Alternatively, $ x $ and 103° are same-side interior angles → $ x + 103 = 180 $ → $ x = 77^\circ $

✔ Answer: $ x = 77^\circ $

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9)

Angles: 114° and 66°, and $ x $ is on the upper line.

Note: 114° + 66° = 180° → they are supplementary.

Now, $ x $ is corresponding to 66°? Or 114°?

Wait — $ x $ is on the upper line, and 66° is on the lower line, on the same side of the transversal → so corresponding angles.

So $ x = 66^\circ $

Alternatively, $ x $ and 114° are same-side interior angles → $ x + 114 = 180 $ → $ x = 66^\circ $

✔ Answer: $ x = 66^\circ $

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✔ Final Answers:

| Problem | Answer |
|--------|--------|
| 1) | $ x = 34^\circ $ |
| 2) | $ x = 93^\circ $ |
| 3) | $ x = 75^\circ $ |
| 4) | $ x = 69^\circ $ |
| 5) | $ x = 30^\circ $ |
| 6) | $ x = 52^\circ $ |
| 7) | $ x = 49^\circ $ |
| 8) | $ x = 77^\circ $ |
| 9) | $ x = 66^\circ $ |

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Summary of Concepts Used:

- Corresponding angles: Equal when lines are parallel.
- Vertical angles: Always equal.
- Linear pairs: Add to 180°.
- Same-side interior angles: Add to 180° when lines are parallel.
- Alternate interior angles: Equal when lines are parallel.

All problems rely on these basic angle relationships.

Let me know if you'd like a visual explanation for any!