Parallel Lines Cut By A Transversal Worksheet, Examples, And Notes - Free Printable

Let's solve each problem step by step using the properties of parallel lines cut by a transversal.

We will use these key angle relationships:

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

---

Problem 1

```
45°
/ \
/ \
/ \
x° y°
```

- The two vertical lines are parallel, and a transversal cuts through them.
- The 45° angle and $ x^\circ $ are vertical angles, so:
$$
x = 45^\circ
$$
- Now, $ x^\circ $ and $ y^\circ $ are alternate interior angles (since they're on opposite sides of the transversal and between the parallel lines), so:
$$
y = x = 45^\circ
$$

✔ Answer:
$ x = 45^\circ $, $ y = 45^\circ $

---

Problem 2

```
60°
/ \
/ \
/ \
x° y°
```

- The two horizontal lines are parallel.
- The 60° angle and $ x^\circ $ are vertical angles, so:
$$
x = 60^\circ
$$
- $ x^\circ $ and $ y^\circ $ are corresponding angles (same position relative to the transversal and parallel lines), so:
$$
y = x = 60^\circ
$$

✔ Answer:
$ x = 60^\circ $, $ y = 60^\circ $

---

Problem 3

```
x°
/ \
/ \
/ \
/ \
119° y°
```

- Parallel lines with a transversal.
- The 119° angle and $ x^\circ $ are vertical angles, so:
$$
x = 119^\circ
$$
- $ x^\circ $ and $ y^\circ $ are alternate interior angles, so:
$$
y = x = 119^\circ
$$

✔ Answer:
$ x = 119^\circ $, $ y = 119^\circ $

---

Problem 4

```
x° y°
/ \ /
/ \
/ \
124°
```

- The two vertical lines are parallel.
- The 124° angle and $ x^\circ $ are supplementary because they form a linear pair (they are adjacent and on a straight line). So:
$$
x + 124^\circ = 180^\circ \Rightarrow x = 56^\circ
$$
- $ x^\circ $ and $ y^\circ $ are vertical angles, so:
$$
y = x = 56^\circ
$$

✔ Answer:
$ x = 56^\circ $, $ y = 56^\circ $

---

Problem 5

```
y°
/ \
/ \
/ \
43° x°
```

- The two vertical lines are parallel.
- The 43° angle and $ x^\circ $ are vertical angles, so:
$$
x = 43^\circ
$$
- $ x^\circ $ and $ y^\circ $ are alternate interior angles, so:
$$
y = x = 43^\circ
$$

✔ Answer:
$ x = 43^\circ $, $ y = 43^\circ $

---

Problem 6

```
x°
/ \
/ \
/ \
y° 151°
```

- Two horizontal parallel lines cut by a transversal.
- The 151° angle and $ y^\circ $ are vertical angles, so:
$$
y = 151^\circ
$$
- $ y^\circ $ and $ x^\circ $ are same-side interior angles (on the same side of the transversal and inside the parallel lines), so they are supplementary:
$$
x + y = 180^\circ \Rightarrow x + 151^\circ = 180^\circ \Rightarrow x = 29^\circ
$$

✔ Answer:
$ x = 29^\circ $, $ y = 151^\circ $

---

✔ Final Answers Summary:

| Problem | x | y |
|--------|---------|---------|
| 1 | 45° | 45° |
| 2 | 60° | 60° |
| 3 | 119° | 119° |
| 4 | 56° | 56° |
| 5 | 43° | 43° |
| 6 | 29° | 151° |

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