How to Prove Lines are Parallel
To prove lines are parallel, you show that a pair of angles made by a transversal obeys one of three rules: alternate angles are equal, corresponding angles are equal, or co-interior angles add up to 180 degrees. This page explains each rule with diagrams and worked examples, then gives you four auto-marked practice rooms with unlimited randomly generated questions.
What does it mean to prove lines are parallel?
Parallel lines stay the same distance apart and never meet, however far they are extended. In an exam you cannot prove lines are parallel by measuring a diagram, because diagrams are often deliberately drawn inaccurately. Instead, you use angles.
A transversal is a straight line that crosses two other lines. When a transversal crosses two lines, it creates eight angles, and three special pairs of those angles behave in a predictable way if and only if the two lines are parallel. So the logic works in both directions:
- If the lines are parallel, the angle rules below must hold. Use this to find missing angles.
- If one of the angle rules holds, the lines must be parallel. Use this to prove lines are parallel.
- If the rule fails, the lines are not parallel.
The three angle facts
| Angle pair | Shape to spot | Rule | Reason to write in an exam |
|---|---|---|---|
| Alternate angles | Z | Equal | "Alternate angles are equal" |
| Corresponding angles | F | Equal | "Corresponding angles are equal" |
| Co-interior angles | C | Add up to 180° | "Co-interior angles add up to 180°" |
The letters Z, F and C are a spotting tool only. In an exam you must write the proper names; "Z angles" scores no marks as a reason.
How to prove two lines are parallel
- Find the transversal: the line that crosses both of the lines in question.
- Pick out a pair of angles in one of the three positions: alternate, corresponding or co-interior.
- Find the size of both angles, using other angle facts if you need to.
- Show that the pair obeys its rule: the angles are equal, or they add up to 180°.
- State the conclusion with the reason, for example: "The alternate angles are equal, so the lines are parallel."
If the pair does not obey its rule, you have proved the opposite: the lines are not parallel. State that instead, with the same style of reason.
Worked examples
💡 Example 1: alternate angles
The two marked angles are both 64°. Prove that the two horizontal lines are parallel.
The marked angles are on opposite sides of the transversal, between the two lines, so they are in alternate positions.
Both angles are 64°, so the alternate angles are equal.
The alternate angles are equal, so the lines are parallel.
What's happening?
Spot the Z shape first. Both marked angles sit inside the Z, so the rule to test is "alternate angles are equal".
The values match, so the conclusion follows. Always finish with the reason written out in full.
💡 Example 2: co-interior angles
The two marked angles are 112° and 68°. Prove that the two horizontal lines are parallel.
The marked angles are on the same side of the transversal, between the two lines, so they are co-interior.
112° + 68° = 180°
The co-interior angles add up to 180°, so the lines are parallel.
What's happening?
This time the two angles form a C shape, so the rule to test is the sum, not equality.
Show the addition explicitly. An examiner wants to see 112 + 68 = 180 on the page, not just the conclusion.
💡 Example 3: showing lines are NOT parallel
The two marked angles are 73° and 105°. Is the top line parallel to the bottom line? Give a reason for your answer.
The marked angles are in alternate positions, so if the lines were parallel the angles would be equal.
73° ≠ 105°
The alternate angles are not equal, so the lines are not parallel.
What's happening?
The diagram is labelled "NOT accurately drawn", so you cannot trust your eyes; the lines look parallel but the angle values decide it.
The method is identical: identify the pair, test the rule, state the conclusion. Here the rule fails, so the answer is no.
🔑 Key Points
- Alternate angles (Z) are equal.
- Corresponding angles (F) are equal.
- Co-interior angles (C) add up to 180°.
- Any one of these three facts is enough to prove two lines are parallel.
- Always state the reason in words; the reason carries marks.
⚠️ Pitfalls
- Never assume lines are parallel because they look parallel; diagrams are often not drawn accurately.
- Do not mix the rules up: co-interior angles add up to 180°, they are not equal.
- Writing "Z angles" or "F angles" as a reason scores nothing; use the full names.
- Check the pair really is between the two lines for alternate and co-interior angles.