Proving Parallel Lines
Proving Parallel Lines turns the angle rules around: instead of using parallel lines to find angles, you use angles to decide whether lines are parallel. In NSW Year 7 maths (NESA outcome MA4‑ANG‑C‑01) this reversed rule is called the converse.
You will test whether corresponding or alternate angles are equal, or whether co-interior angles add to \(180^\circ\), then write a conclusion with its reason. You will also find the value that would make two lines parallel and judge an argument about two roads crossed by a highway. This subtopic is part of Year 7 Angles.
Every question with a fully worked solution.
- Proving Parallel Lines - Video - Proving Lines are Parallel Watch
Theory
To prove two lines are parallel, show that a pair of corresponding angles are equal, a pair of alternate angles are equal, or a pair of co-interior angles add to \(180^\circ\). This Year 7 maths page sets out the three steps and works through parallel, not-parallel and two-step proofs.
Until now the arrow marks told you the lines were parallel, and you found the angles. Now the angles tell you whether the lines are parallel. This is the converse of each rule.
Two lines are parallel if alternate angles are equal, or corresponding angles are equal, or co-interior angles are supplementary (add to \(180^\circ\)).
Three steps. (1) Locate the two angles and name the pair. (2) Test: are they equal, or do they add to \(180^\circ\)? (3) Write the conclusion with its reason.
If the test fails, the lines are not parallel: “The co-interior angles add to \(175^\circ\), not \(180^\circ\), so \(AB\) is not parallel to \(CD\).”
If ..., then write ...:
| If | Then write |
|---|---|
| corresponding angles are equal | The corresponding angles are equal, so \(AB \parallel CD\). |
| alternate angles are equal | The alternate angles are equal, so \(AB \parallel CD\). |
| co-interior angles add to \(180^\circ\) | The co-interior angles are supplementary, so \(AB \parallel CD\). |
How to prove two lines are parallel
- Locate the two angles and name the pair (corresponding, alternate or co-interior).
- Test: equal (corresponding, alternate) or adding to \(180^\circ\) (co-interior)?
- Conclude with the reason, or say the lines are not parallel.
Both angles are between the lines, on opposite sides: alternate angles. Test: \(73^\circ = 73^\circ\).
Yes. The alternate angles are equal, so \(AB \parallel CD\).
Both angles are between the lines, on the same side: co-interior angles.
| \(96 + 81\) | \(=\) | \(177\) |
No. \(177 \ne 180\): the co-interior angles are not supplementary, so \(AB\) is not parallel to \(CD\).
Name each angle with its vertex in the middle: the vertex is the crossing point (\(K\) or \(L\)), not the end of the transversal. \(\angle QKL\) and \(\angle KLS\) are co-interior.
| \(100 + 85\) | \(=\) | \(185\) |
\(185 \ne 180\), so \(PQ\) is not parallel to \(RS\).
The angles are alternate. For \(AB \parallel CD\) they must be equal.
| \(x + 20\) | \(=\) | \(75\) |
| \(x\) | \(=\) | \(55\) |
\(x = 55\). Check: \(55 + 20 = 75\).
\(\angle EGB\) and \(\angle CHF\) are not a pair yet. Vertically opposite angles are equal: \(\angle AGH = \angle EGB = 68^\circ\). \(\angle AGH\) and \(\angle CHF\) are both below their line, left of \(EF\): corresponding angles.
\(\angle AGH = \angle CHF = 68^\circ\). The corresponding angles are equal, so \(AB \parallel CD\).
(ii) Quadrilateral \(ABCD\) has \(\angle A = 55^\circ\), \(\angle B = 120^\circ\), \(\angle C = 60^\circ\) and \(\angle D = 125^\circ\). Which sides are parallel?
| \(58 + 122\) | \(=\) | \(180\) |
| \(55 + 125\) | \(=\) | \(180\) |
| \(55 + 120\) | \(=\) | \(175\) |
(i) Yes: the co-interior angles are supplementary.
(ii) \(AD\) is a transversal for \(AB\) and \(DC\): \(\angle A + \angle D = 180^\circ\), so \(AB \parallel DC\). \(\angle A + \angle B = 175^\circ\), so \(AD\) is not parallel to \(BC\). Only \(AB \parallel DC\).
Common pitfalls
Frequently asked questions
How do you prove two lines are parallel?
Show corresponding angles are equal, alternate angles are equal, or co-interior angles add to 180°.
What does converse mean here?
The angle rule used backwards: the angles show the lines are parallel.
Can vertically opposite angles prove lines are parallel?
No. They are always equal.
How do you show lines are not parallel?
Show a pair of alternate or corresponding angles are not equal, or co-interior angles do not add to \(180^\circ\).
What value makes two lines parallel?
The value that makes the alternate or corresponding angles equal, or the co-interior angles add to 180°.