Angles on Parallel Lines
When a transversal crosses two parallel lines it makes eight angles but only two different sizes. In NSW Year 8 maths (NESA outcome MA4‑ANG‑C‑01) corresponding and alternate angles are equal and co-interior angles add to \(180^\circ\), and each step names the pair used and the parallel lines, such as \(AB \parallel CD\).
You will find unknown angles in diagrams with one or two sets of parallel lines, solve for angles written as expressions such as \(3a\), and draw a construction line through a bend to split the angle there. You will also use angle pairs to decide whether two lines are parallel, check another student's reasons, and work with sunlight on flagpoles and the lines of a car park. It is part of Year 8 Geometry.
Theory
A transversal across two parallel lines makes eight angles of at most two sizes. Corresponding angles are equal, alternate angles are equal, and co-interior angles add to \(180^\circ\). Name the pair and the parallel lines in every reason. This Year 8 maths page shows how.
A transversal is a line that crosses two or more lines. Arrow marks show which lines are parallel; \(AB \parallel CD\) is read “\(AB\) is parallel to \(CD\)”.
Corresponding angles (F shape) sit in the same position at each crossing. On parallel lines they are equal.
Alternate angles (Z shape) lie between the lines, on opposite sides of the transversal. On parallel lines they are equal.
Co-interior angles (C shape) lie between the lines, on the same side of the transversal. On parallel lines they are supplementary: they add to \(180^\circ\).
The reason names the lines: write (alternate angles, \(AB \parallel CD\)), not just “parallel lines”. The other angle facts still apply.
Two sets of parallel lines. Single arrows mark one pair, double arrows the other. Check which pair the transversal crosses before you choose a reason.
Going the other way. If corresponding or alternate angles are equal, or co-interior angles add to \(180^\circ\), then the lines are parallel.
A bend between the lines. Draw a construction line through the bend, parallel to both lines. It splits the angle at the bend into two alternate angles; add them.
On parallel lines:
| Pair | Shape | Rule |
|---|---|---|
| corresponding | F | equal |
| alternate | Z | equal |
| co-interior | C | add to \(180^\circ\) |
| vertically opposite | X | equal (any crossing lines) |
How to find an angle on parallel lines
- Find the transversal and the parallel lines it crosses.
- Locate the two angles: between the lines or outside, same side or opposite sides.
- Name the pair and use its rule, with the reason (pair, \(AB \parallel CD\)).
\(x^\circ\) and \(136^\circ\) are between the lines on the same side: co-interior. \(y^\circ\) is in the same position at \(CD\) as \(136^\circ\) is at \(AB\): corresponding.
| \(x + 136\) | \(=\) | \(180\) |
| \(x\) | \(=\) | \(44\) |
| \(y\) | \(=\) | \(136\) |
\(x = 44\) (co-interior angles, \(AB \parallel CD\)) and \(y = 136\) (corresponding angles, \(AB \parallel CD\)).
| \((2x + 3) + (x + 12)\) | \(=\) | \(180\) |
| \(3x + 15\) | \(=\) | \(180\) |
| \(3x\) | \(=\) | \(165\) |
| \(x\) | \(=\) | \(55\) |
\(x = 55\); the angles are \(2 \times 55 + 3 = 113^\circ\) and \(55 + 12 = 67^\circ\). Check: \(113 + 67 = 180\).
| \(148 + 32\) | \(=\) | \(180\) |
Yes: \(AB \parallel CD\), because the co-interior angles add to \(180^\circ\). Had the second angle been \(36^\circ\), the total would be \(184\) and the lines would not be parallel.
Draw \(XY\) through \(X\), parallel to \(AB\) and \(CD\). Then \(\angle PXY = 29^\circ\) (alternate angles, \(AB \parallel XY\)) and \(\angle YXQ = 43^\circ\) (alternate angles, \(CD \parallel XY\)).
| \(\angle PXQ\) | \(=\) | \(29 + 43 = 72^\circ\) |
| \(x + 72\) | \(=\) | \(360\) |
| \(x\) | \(=\) | \(288\) |
\(x = 288\) (angles at a point).
(i) the angle below \(AB\), left of the transversal
(ii) the angle above \(CD\), right of the transversal
(iii) the angle above \(CD\), left of the transversal
(i) \(80^\circ\) (vertically opposite angles)
(ii) \(80^\circ\) (corresponding angles, \(AB \parallel CD\))
(iii) \(180 - 80 = 100^\circ\) (angles on a straight line)
Common pitfalls
Frequently asked questions
What are corresponding angles?
Angles in the same position at each crossing of a transversal. On parallel lines they are equal.
What are alternate angles?
Angles between the parallel lines on opposite sides of the transversal, in a Z shape. They are equal.
What are co-interior angles?
Angles between the parallel lines on the same side of the transversal, in a C shape. They add to \(180^\circ\).
How do I prove two lines are parallel?
Show that a pair of alternate or corresponding angles is equal, or that co-interior angles add to \(180^\circ\).
What is a construction line?
An extra line you draw, such as a line through a bend parallel to the given lines, to create angle pairs you can use.