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 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.

---

1)

```
←───────────→
146° 34°
\ /
\ /
x
```

The angle marked 34° is adjacent to x, and together they form a straight line with the 146° angle.

But wait — let’s analyze carefully.

Actually, the two parallel lines are horizontal. A transversal cuts them forming angles.

From the diagram:
- The top angle on the upper line is 146° and 34°.
- Since 146° + 34° = 180°, they are on a straight line → so the transversal makes a straight angle.

Now, x is an angle on the lower line, corresponding to the 34° angle.

Since the lines are parallel, corresponding angles are equal.

So, x = 34°

✔ Answer: x = 34°

---

2)

```
↑ ↑
| |
| 93° |
| |
↓ ↓
←────→
87°
x
```

Wait — this looks like a transversal cutting two vertical parallel lines? But actually, the arrows suggest horizontal lines.

Looking at it again:

Two horizontal parallel lines, a vertical transversal.

At the top intersection:
- One angle is 93°, and another is 87°.

Wait — 93° + 87° = 180° → they are on a straight line.

So, the angle adjacent to 93° is 87°, and vice versa.

Now, x is below, on the lower line, opposite the 87° angle.

But x is directly across from the 87° angle? Or is it?

Wait — in the diagram, x is labeled at the bottom, and there's an 87° angle above it.

If the transversal is vertical, and the lines are horizontal, then:

- The angle x and the 87° angle are vertical angles (opposite angles formed by intersecting lines).

Vertical angles are equal → so x = 87°

Alternatively, if you consider corresponding angles: the angle x corresponds to the 87° angle above it? Yes — same position relative to the transversal and the lines.

So yes, x = 87°

✔ Answer: x = 87°

---

3)

```
←───────────→
75° 105°
\ /
\ /
x
```

Top line has two angles: 75° and 105°. They are on a straight line → 75° + 105° = 180° → correct.

Now, the transversal goes down to the lower line, where we have angle x.

Angle x is corresponding to the 75° angle (same side, same relative position).

So, since lines are parallel, x = 75°

✔ Answer: x = 75°

---

4)

```
↑ ↑
| |
| x |
| |
↓ ↓
←────→
111°
```

Wait — no, the diagram shows:

A transversal crossing two parallel lines.

On the top line, the angle is 111°, and on the bottom line, x is labeled.

But x is on the same side as the 111° angle, but on the other side of the transversal?

Wait — looking at the diagram:

- The 111° is on the upper right.
- x is on the lower left.

They are not corresponding or alternate.

But notice: 111° and x are alternate interior angles?

Wait — let's think.

The transversal crosses two parallel lines.

The 111° angle is on the top line, on the right side of the transversal.

Then, x is on the bottom line, on the left side of the transversal.

That would make them alternate interior angles if they're between the lines.

But here, x appears to be on the outside.

Wait — perhaps the 111° is on the top, and x is on the bottom, both on the same side of the transversal.

Wait — actually, the diagram shows:

- The 111° is on the top, right side.
- x is on the bottom, left side.

But they are not corresponding.

Wait — maybe x is vertical to some angle?

Let’s reconstruct:

The 111° is formed at the top line. Its adjacent angle (on the same line) is 180° - 111° = 69°.

Now, that 69° is corresponding to x?

Wait — no.

Wait — perhaps x is corresponding to the 111° angle?

No — because x is on the opposite side of the transversal.

Wait — look again.

In standard diagrams, if the transversal slants from top-left to bottom-right:

- The 111° is on the top, right.
- Then, the corresponding angle on the bottom would be on the bottom, right — but x is on the left.

So, x is not corresponding.

Wait — maybe x is vertical to an angle?

Wait — perhaps the 111° and x are alternate interior?

No — alternate interior would be inside the lines, on opposite sides.

But x appears to be outside.

Wait — actually, x is on the same side of the transversal as the 111°, but on the other line.

So, they are corresponding angles?

Yes! If the transversal goes from top-left to bottom-right, and the lines are horizontal:

- Top line: angle on the right side is 111°
- Bottom line: angle on the right side is x

Wait — but in the diagram, x is labeled on the left side.

Wait — perhaps I'm misreading.

Let me interpret carefully:

The diagram shows:

- Two vertical arrows (parallel lines)
- A diagonal transversal going from bottom-left to top-right

At the top intersection:
- The angle between the transversal and the top line is 111°, on the right side

At the bottom intersection:
- The angle x is on the left side

So, x is not corresponding to 111°.

But 111° and x are vertical angles?

No — they’re not at the same vertex.

Wait — perhaps x is corresponding to the supplement of 111°?

Let’s do this:

At the top, the angle is 111°, so its adjacent angle (on the same line) is 180° - 111° = 69°

Now, that 69° is on the top, left side.

Then, the corresponding angle on the bottom line would be on the bottom, left side — which is x

So, x = 69°

✔ Answer: x = 69°

---

5)

```
←───────────→
x
/ \
/ \
/ \
←───────→
150° 30°
```

Wait — two horizontal parallel lines.

Top line has angle x.

Bottom line has 150° and 30°.

Wait — 150° + 30° = 180° → so they are adjacent angles on a straight line.

So, the transversal forms a 150° angle on the bottom, and a 30° angle on the bottom.

Now, x is on the top line, and it's corresponding to the 30° angle?

Wait — the 30° is on the bottom, right side.

Then, x is on the top, left side?

Wait — no — x is on the top, and the 30° is on the bottom.

But they are on opposite sides.

Wait — perhaps x is corresponding to the 150°?

No — 150° is large, x might be small.

Wait — let's think.

The 30° angle on the bottom is adjacent to the 150°.

Now, the 30° angle is between the transversal and the bottom line.

Its corresponding angle on the top line would be in the same position.

So, if the transversal comes from bottom-left to top-right, and the 30° is on the bottom, right, then the corresponding angle on the top, right would be 30°.

But x is labeled on the top, left?

Wait — the diagram shows:

- On the top line, the angle x is on the left side of the transversal.
- On the bottom line, 150° is on the left, 30° on the right.

So, x is on the top, left → so it's corresponding to the 150° angle?

Yes!

Because both are on the left side of the transversal, and on the same relative side of the lines.

So, x = 150°

Wait — but 150° is obtuse, and x seems acute?

Wait — perhaps x is alternate interior?

Wait — let's use vertical angles.

At the bottom, the 30° angle is adjacent to the 150°.

The angle vertically opposite to 30° is also 30°.

But x is on the top.

Wait — perhaps x is corresponding to the 30°?

But 30° is on the right, and x is on the left.

No.

Wait — perhaps x is equal to 30° because it's alternate interior?

Let’s see:

The 30° angle is on the bottom, right.

The x is on the top, left.

Not matching.

Wait — what about same-side interior?

Wait — perhaps x and 150° are corresponding?

Yes — if x is on the top, left, and 150° is on the bottom, left, then they are corresponding angles → so x = 150°

But that seems large.

Wait — perhaps the 30° is corresponding to x?

But 30° is on the right, x on the left — not same side.

Wait — unless the transversal is different.

Wait — maybe the 30° and x are vertical angles?

No — different vertices.

Wait — perhaps x is supplementary to something.

Another idea:

At the bottom, the 30° angle is formed.

Its vertical angle is also 30°.

That 30° is on the bottom, left side? No — vertical angle is opposite.

Wait — the 30° is on the bottom, right.

Its vertical angle is on the bottom, left — but that's the 150°? No — 150° + 30° = 180° → so they are adjacent.

Wait — actually, the 30° and 150° are adjacent angles on the bottom line → so they are linear pair.

So, the angle opposite to 30° is 30°, and opposite to 150° is 150°.

But x is on the top line.

Now, the angle on the top line, right side, should be equal to the 30° on the bottom (corresponding).

But x is on the left side.

So, x is on the top, left — so it should correspond to the 150° on the bottom, left.

So, x = 150°

But that seems odd — let’s check.

Yes — x is corresponding to the 150° angle → so x = 150°

✔ Answer: x = 150°

Wait — but in many such problems, the smaller angle is used.

But mathematically, if the lines are parallel, corresponding angles are equal.

So, x = 150°

But let’s double-check.

Alternative: x and 30° are alternate interior?

No — not on opposite sides.

Wait — perhaps x is supplementary to 30°?

No.

Wait — maybe x is equal to 30°?

Let’s assume the transversal creates 30° on the bottom, right.

Then, the corresponding angle on the top, right is 30°.

But x is on the top, left.

So, the angle adjacent to x on the top line would be 30°, so x = 180° - 30° = 150°

Ah! That’s it.

So, on the top line, the angle adjacent to x is 30° (corresponding to the 30° on the bottom), so x is supplementary to it.

So, x = 180° - 30° = 150°

✔ Answer: x = 150°

---

6)

```
↑ ↑
| |
| x |
| |
↓ ↓
←────→
52°
```

Two vertical parallel lines, a diagonal transversal.

At the bottom, the angle is 52°.

At the top, x is labeled.

Now, the 52° angle is on the bottom, right side.

Its vertical angle is also 52°.

Now, x is on the top, left side.

Is x corresponding to the 52°?

No — 52° is on the right, x on the left.

But wait — x is on the top, left, and 52° is on the bottom, right.

They are not corresponding.

Wait — perhaps x is alternate interior?

No.

Wait — the 52° angle is on the bottom, right.

Its corresponding angle on the top, right would be 52°.

But x is on the top, left.

So, x is vertical to the corresponding angle?

Wait — the corresponding angle to 52° is 52° on the top, right.

Then, x is vertical to that angle?

No — x is on the top, left.

Wait — perhaps x is equal to 52° because it's corresponding to the 52°?

No — not same side.

Wait — perhaps x and 52° are alternate interior?

No — alternate interior would be between the lines, on opposite sides.

But here, the lines are vertical, so "inside" is horizontal.

Wait — let’s think.

The 52° is on the bottom, right.

Its corresponding angle on the top, right is 52°.

Then, x is on the top, left — so it's adjacent to that 52° angle.

So, x and 52° are on a straight line → x + 52° = 180°

So, x = 180° - 52° = 128°

✔ Answer: x = 128°

---

7)

```
←───────────→
x
/ \
/ \
/ \
←───────→
131° 49°
```

Two parallel lines, transversal.

On the bottom line: 131° and 49°

131° + 49° = 180° → good.

Now, x is on the top line.

It’s on the left side.

The 131° is on the bottom, left.

So, x is corresponding to 131° → so x = 131°

✔ Answer: x = 131°

---

8)

```
↑ ↑
| |
| x |
| |
↓ ↓
←────→
103°
77°
```

Two horizontal parallel lines, vertical transversal.

At the bottom: 103° and 77°

103° + 77° = 180° → good.

Now, x is on the top, left.

The 77° is on the bottom, left.

So, x is corresponding to 77° → so x = 77°

✔ Answer: x = 77°

---

9)

```
←───────────→
x
/ \
/ \
/ \
←───────→
114° 66°
```

Two parallel lines, transversal.

On the bottom: 114° and 66°

114° + 66° = 180° → good.

Now, x is on the top, left.

The 114° is on the bottom, left.

So, x is corresponding to 114° → so x = 114°

✔ Answer: x = 114°

---

## ✔ Final Answers:

1) x = 34°
2) x = 87°
3) x = 75°
4) x = 69°
5) x = 150°
6) x = 128°
7) x = 131°
8) x = 77°
9) x = 114°

---

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