# Angles and Parallel Lines

> Spot corresponding, alternate, co-interior and vertically opposite angles, and decide if lines are parallel.

From Selective School Maths on WhizzNotes, free study notes for NSW students: https://whizznotes.com/selective-school-maths/angles-and-parallel-lines/. Updated 3 November 2026.
Year 11 and Year 12 boxes go deeper than the Year 10 text around them.

### Story time: The railway crossing
Mei's family lives near a train line. A straight road crosses the two parallel rails at a slant.

Looking down from the footbridge, Mei notices something. The angle the road makes with the first rail looks exactly the same as the angle it makes with the second rail. "Of course," says her dad, an engineer. "The rails point the same way, so the road meets them both the same way."

That simple idea gives us a set of angle rules that appear on nearly every selective test.

**In short:** When a line crosses two parallel lines, it makes angles that follow three rules. Corresponding angles (an F shape) are equal, alternate angles (a Z shape) are equal, and co-interior angles (a C shape) add to $180^\circ$.

## Before you start

- **Parallel lines** never meet. Diagrams show them with matching arrows.
- A line that cuts across two or more lines is called a **transversal**.
- Angles on a straight line add to $180^\circ$. Vertically opposite angles are equal (see Types of Angles).

## Eight angles, four rules

When a transversal crosses two parallel lines, it makes 8 angles: 4 at each crossing.

*Diagram: Two parallel lines cut by a slanted transversal; angles a, b, c, d at the top crossing and e, f, g, h at the bottom crossing, in the same positions*
*Angles in the same position at each crossing match: a and e, b and f, c and g, d and h.*

### Key idea: Corresponding angles are equal (an F shape)
Angles in the **same position** at each crossing are equal: $a = e$, $b = f$, $c = g$, $d = h$.

Trace an F (forwards or backwards, upside down or not) and the angles under the arms of the F are corresponding.

### Key idea: Alternate angles are equal (a Z shape)
Angles **between** the parallel lines, on **opposite sides** of the transversal, are equal: $c = f$ and $d = e$.

Trace a Z (or a backwards Z) and the angles in its corners are alternate.

### Key idea: Co-interior angles add to $180^\circ$ (a C shape)
Angles **between** the parallel lines, on the **same side** of the transversal, add up to $180^\circ$: $c + e = 180^\circ$ and $d + f = 180^\circ$.

Trace a C (or a backwards C). Co-interior angles are **supplementary**.

### Key idea: Vertically opposite angles are equal (an X shape)
At each crossing, opposite angles are equal: $a = d$, $b = c$, $e = h$, $f = g$.

## Solving with the rules

### Worked example: One angle gives all eight
In the diagram, $b = 120^\circ$.

$a = 180 - 120 = 60^\circ$ (straight line). $c = b = 120^\circ$ (vertically opposite). $d = a = 60^\circ$.

At the bottom crossing every angle copies the top one: $e = 60^\circ$, $f = 120^\circ$, $g = 120^\circ$, $h = 60^\circ$.

### Exam tip: Only two sizes
With two parallel lines and one transversal there are only **two** different angle sizes, and they add to $180^\circ$. Every acute-looking angle is the small one, and every obtuse-looking angle is the big one.

### Watch out: The lines must be parallel
These rules only work when the lines are parallel (look for the arrows). If the lines are not parallel, corresponding angles are **not** equal.

## Are the lines parallel?

You can use the rules backwards.

- If corresponding angles are equal, the lines are parallel.
- If alternate angles are equal, the lines are parallel.
- If co-interior angles add to $180^\circ$, the lines are parallel.

### Worked example: Test it
A transversal makes a $70^\circ$ angle with one line and the co-interior angle on the other line is $120^\circ$. Since $70 + 120 = 190$, not 180, the lines are **not** parallel.

### Challenge: Selective level: angles with two steps
A transversal crosses two parallel lines. An angle at the top crossing is $105^\circ$. Find the alternate angle $h$ and the angle $i$ next to it on the straight line: $h = 105^\circ$ and $i = 180 - 105 = 75^\circ$.

Two parallel lines are cut by a transversal. Two co-interior angles are $(2x + 10)^\circ$ and $(3x - 30)^\circ$. They add to 180: $5x - 20 = 180$, so $x = 40$, giving $90^\circ$ and $90^\circ$. The transversal is at right angles to both lines.

**Mind map**

```text
Parallel line angles
  Corresponding
    F shape
    Equal
  Alternate
    Z shape
    Equal
  Co-interior
    C shape
    Add to 180°
  Vertically opposite
    X shape
    Equal
  Parallel test
    Use the rules backwards
```

## Quick summary

- A transversal crossing two parallel lines makes 8 angles with only 2 different sizes.
- Corresponding angles (F) are equal.
- Alternate angles (Z) are equal.
- Co-interior angles (C) add to $180^\circ$.
- Vertically opposite angles (X) are equal.
- Use the rules backwards to show lines are parallel.

## Quiz

Try these questions from the chapter quiz.

1. Corresponding angles between parallel lines are:
   - A. equal
   - B. add to $90^\circ$
   - C. add to $180^\circ$
   - D. add to $360^\circ$
   - Answer: A. equal. Corresponding angles (the F shape) are equal.

2. Co-interior angles between parallel lines:
   - A. are equal
   - B. add to $90^\circ$
   - C. add to $180^\circ$
   - D. add to $360^\circ$
   - Answer: C. add to $180^\circ$. Co-interior angles (the C shape) add to $180^\circ$.

3. Which letter shape shows alternate angles?
   - A. F
   - B. Z
   - C. C
   - D. X
   - Answer: B. Z. Alternate angles sit in the corners of a Z.

The whole chapter quiz is on the page, for signed-in students: https://whizznotes.com/selective-school-maths/angles-and-parallel-lines/#quiz