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. The key angle relationships we'll use are:

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

---

Problem 1

```
↑ ↑
45° x°
\ /
\ /
\/
y°
```

- The 45° angle and $x^\circ$ are vertical angles, so:
$$
x = 45^\circ
$$
- $x^\circ$ and $y^\circ$ form a linear pair (they are on a straight line), so:
$$
x + y = 180^\circ \Rightarrow 45 + y = 180 \Rightarrow y = 135^\circ
$$

✔ Answer: $x = 45^\circ$, $y = 135^\circ$

---

Problem 2

```
<----> <---->
\ /
\ /
\ /
x°
y°
```

Given: One angle is $60^\circ$, and the lines are parallel.

- $x^\circ$ and $60^\circ$ are vertical angles, so:
$$
x = 60^\circ
$$
- $y^\circ$ and $60^\circ$ are alternate interior angles (since lines are parallel), so:
$$
y = 60^\circ
$$

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

---

Problem 3

```
<----> <---->
\ /
\ /
\ /
x°
y°
119°
```

The given angle is $119^\circ$. This is between the lower line and the transversal.

- $x^\circ$ and $119^\circ$ are vertical angles, so:
$$
x = 119^\circ
$$
- $y^\circ$ and $119^\circ$ are same-side interior angles (on the same side of the transversal, inside the parallel lines). They are supplementary:
$$
y + 119 = 180 \Rightarrow y = 61^\circ
$$

✔ Answer: $x = 119^\circ$, $y = 61^\circ$

---

Problem 4

```
↑ ↑
x° y°
/ \
/ \
/ \
124°
```

We have a transversal cutting two vertical parallel lines.

- $x^\circ$ and $124^\circ$ are vertical angles, so:
$$
x = 124^\circ
$$
- $y^\circ$ and $124^\circ$ are same-side interior angles → they are supplementary:
$$
y + 124 = 180 \Rightarrow y = 56^\circ
$$

✔ Answer: $x = 124^\circ$, $y = 56^\circ$

---

Problem 5

```
↑ ↑
y° x°
/ \
/ \
/ \
43°
```

- $x^\circ$ and $43^\circ$ are vertical angles, so:
$$
x = 43^\circ
$$
- $y^\circ$ and $43^\circ$ are corresponding angles (or alternate interior), so since lines are parallel:
$$
y = 43^\circ
$$

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

---

Problem 6

```
<----> <---->
\ /
\ /
\ /
x°
y°
151°
```

- The $151^\circ$ angle and $x^\circ$ are vertical angles, so:
$$
x = 151^\circ
$$
- $y^\circ$ and $151^\circ$ are same-side interior angles (between the parallel lines, on the same side of the transversal) → supplementary:
$$
y + 151 = 180 \Rightarrow y = 29^\circ
$$

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

---

✔ Final Answers Summary:

| Problem | $x$ | $y$ |
|--------|-----|-----|
| 1 | 45° | 135° |
| 2 | 60° | 60° |
| 3 | 119°| 61° |
| 4 | 124°| 56° |
| 5 | 43° | 43° |
| 6 | 151°| 29° |

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