Find the value of x in each figure where two parallel lines are cut by a transversal.

Let's solve each of these problems step by step using the properties of angles formed when two parallel lines are cut by a transversal.

Key Concepts:

- Alternate interior angles are equal.
- Alternate exterior angles are equal.
- Corresponding angles are equal.
- Supplementary angles (on a straight line) add up to 180°.
- Vertical angles are equal.

We’ll use these rules to find $ x $ in each figure.

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1)

Given:
- One angle is $ 144^\circ $, and an adjacent angle is $ 36^\circ $.
- The lines are parallel, and a transversal cuts them.

We see that $ 144^\circ + 36^\circ = 180^\circ $, so they form a linear pair.
The angle marked $ x $ is alternate interior to the $ 36^\circ $ angle.

So, since alternate interior angles are equal:

$$
x = 36^\circ
$$

✔ Answer: $ x = 36^\circ $

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2)

Angles given:
- $ 131^\circ $ and $ 49^\circ $ on one side, and $ x $ opposite to $ 49^\circ $.

Note: $ 131^\circ + 49^\circ = 180^\circ $, so they're supplementary — likely on a straight line.

But $ x $ is vertically opposite to $ 49^\circ $? Wait — let’s look carefully.

Actually, $ x $ appears to be corresponding or alternate to $ 49^\circ $, but more clearly, it's vertical to the angle adjacent to $ 131^\circ $.

Wait — better: The angle adjacent to $ 131^\circ $ is $ 180^\circ - 131^\circ = 49^\circ $, which matches the other angle.

Now, $ x $ is vertically opposite to $ 49^\circ $, so:

$$
x = 49^\circ
$$

Alternatively, $ x $ is alternate interior to $ 49^\circ $, so same result.

✔ Answer: $ x = 49^\circ $

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3)

Given:
- $ 127^\circ $ and $ 53^\circ $, and $ x $ is on the lower line.

Check: $ 127^\circ + 53^\circ = 180^\circ $, so they’re supplementary — likely forming a straight line.

Now, $ x $ is alternate interior to $ 53^\circ $, so:

$$
x = 53^\circ
$$

✔ Answer: $ x = 53^\circ $

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4)

Given:
- Two angles on the top line: $ 62^\circ $ and $ 118^\circ $, and $ x $ below.

Note: $ 62^\circ + 118^\circ = 180^\circ $, so they’re supplementary — likely on a straight line.

Now, $ x $ is alternate interior to $ 62^\circ $, because it's on the opposite side of the transversal and between the parallel lines.

So:

$$
x = 62^\circ
$$

✔ Answer: $ x = 62^\circ $

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5)

Given:
- One angle is $ 125^\circ $, another is $ 55^\circ $, and $ x $ is between the lines.

Note: $ 125^\circ + 55^\circ = 180^\circ $, so they are supplementary.

$ x $ is alternate interior to $ 55^\circ $, so:

$$
x = 55^\circ
$$

✔ Answer: $ x = 55^\circ $

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6)

Given:
- Angles $ 102^\circ $ and $ 78^\circ $, and $ x $ is between them.

Note: $ 102^\circ + 78^\circ = 180^\circ $, so they are supplementary.

Now, $ x $ is alternate interior to $ 78^\circ $, so:

$$
x = 78^\circ
$$

✔ Answer: $ x = 78^\circ $

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7)

Given:
- $ 151^\circ $ and $ 29^\circ $, and $ x $ is on the bottom line.

Note: $ 151^\circ + 29^\circ = 180^\circ $, so they are supplementary.

Now, $ x $ is alternate interior to $ 29^\circ $, so:

$$
x = 29^\circ
$$

✔ Answer: $ x = 29^\circ $

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8)

Given:
- Two angles: $ 83^\circ $ and $ 97^\circ $, and $ x $ above.

First, check if they add to 180°: $ 83^\circ + 97^\circ = 180^\circ $, yes — so they’re supplementary.

Now, $ x $ is corresponding to $ 83^\circ $? Or alternate?

Looking at the diagram: $ x $ is on the upper line, and $ 83^\circ $ is on the lower line, same side — possibly corresponding.

But wait: $ x $ is adjacent to $ 83^\circ $ on the same side? No — actually, $ x $ is above the transversal, and $ 83^\circ $ is below.

But notice: $ 83^\circ $ and $ 97^\circ $ are on the same side of the transversal.

Wait — perhaps $ x $ is vertical to the angle adjacent to $ 83^\circ $?

Better: Since the lines are parallel, and $ 83^\circ $ and $ 97^\circ $ are on the same side, then $ x $ is supplementary to $ 83^\circ $?

No — actually, the angle adjacent to $ 83^\circ $ is $ 97^\circ $, and $ x $ is corresponding to $ 83^\circ $, so:

$$
x = 83^\circ
$$

Yes — $ x $ is corresponding to $ 83^\circ $, so equal.

✔ Answer: $ x = 83^\circ $

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9)

Given:
- $ 130^\circ $ and $ 50^\circ $, and $ x $ is on the upper right.

Note: $ 130^\circ + 50^\circ = 180^\circ $, so supplementary.

Now, $ x $ is alternate interior to $ 50^\circ $, so:

$$
x = 50^\circ
$$

✔ Answer: $ x = 50^\circ $

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✔ Final Answers:

| Problem | $ x $ |
|--------|--------|
| 1) | $ 36^\circ $ |
| 2) | $ 49^\circ $ |
| 3) | $ 53^\circ $ |
| 4) | $ 62^\circ $ |
| 5) | $ 55^\circ $ |
| 6) | $ 78^\circ $ |
| 7) | $ 29^\circ $ |
| 8) | $ 83^\circ $ |
| 9) | $ 50^\circ $ |

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Let me know if you'd like diagrams drawn or explanations for any specific one!