Parallel Lines
Every question with a fully worked solution.
- Parallel Lines - Video - Find the Angle Measure Watch
Theory
When a transversal cuts two parallel lines, corresponding angles are equal, alternate angles are equal and co-interior angles add to 180°. This Year 9 maths page revises the Year 7 parallel-line facts: naming the angle pairs, angles written with pronumerals, giving reasons, testing whether two lines are parallel, and using parallel lines inside triangles and at a bend.
A transversal is a line that cuts two or more other lines. Parallel lines never meet; on a diagram they are marked with matching arrows, and in writing with \(\parallel\), as in \(AB \parallel CD\).
When a transversal cuts two parallel lines it makes three special pairs of angles. Corresponding angles (an F shape) are equal. Alternate angles (a Z shape) are equal. Co-interior angles (a C shape, both between the lines on the same side) add to \(180^\circ\).
Some facts hold for any lines: angles on a straight line add to \(180^\circ\), angles at a point add to \(360^\circ\), and vertically opposite angles are equal.
The facts also work backwards. If a pair of corresponding or alternate angles are equal, or a pair of co-interior angles add to \(180^\circ\), the two lines are parallel. If not, they are not parallel.
Always give a reason with each step, such as "alternate angles, \(AB \parallel CD\)".
For a transversal cutting two parallel lines:
| Fact | Rule | Reason to write |
|---|---|---|
| Straight line | add to \(180^\circ\) | angles on a straight line |
| At a point | add to \(360^\circ\) | angles at a point |
| X crossing | equal | vertically opposite angles |
| F shape | equal | corresponding angles, \(AB \parallel CD\) |
| Z shape | equal | alternate angles, \(AB \parallel CD\) |
| C shape | add to \(180^\circ\) | co-interior angles, \(AB \parallel CD\) |
| Triangle | add to \(180^\circ\) | angle sum of a triangle |
How to find an unknown angle
- Mark every angle you know on the diagram, and the parallel-line arrows.
- Look for a shape that joins the unknown to a known angle: an F, Z or C on parallel lines, a straight line, a point or a triangle.
- Find one angle at a time, writing the reason next to it.
- Form an equation if the angles are written with a pronumeral, then solve it and substitute back.
- Check the angles fit: a straight line still adds to \(180^\circ\), a triangle to \(180^\circ\).
Useful constructions and checks
- A bend between parallel lines: draw a third parallel line through the corner and split the angle in two (Example 4). A reflex angle at the bend is \(360^\circ\) minus the angle inside.
- A line parallel to one side of a triangle: it makes corresponding angles with the other two sides (Example 3).
- A trapezium: its parallel sides make co-interior angles that add to \(180^\circ\), and a diagonal makes alternate angles.
- An isosceles triangle between the lines: its base angles are equal, so find one from the parallel lines and copy it to the other, then use the angle sum.
- Checking someone's reasons: F means corresponding, Z means alternate, C means co-interior. A pair of equal angles called co-interior is a wrong reason.
\(a\) sits in the same position at the other intersection: an F shape.
| \(a\) | \(=\) | \(\mathbf{63^\circ}\) |
| \(\text{(corresponding, parallel lines)}\) |
\(a\) and \(b\) make a straight line.
| \(b\) | \(=\) | \(180^\circ - 63^\circ\) |
| \(\) | \(=\) | \(\mathbf{117^\circ}\ \text{(straight angle)}\) |
\(c\) and the \(63^\circ\) angle are opposite each other at a crossing.
| \(c\) | \(=\) | \(\mathbf{63^\circ}\ \text{(vertically opposite)}\) |
Check: \(b\) and \(c\) are co-interior and \(117^\circ + 63^\circ = 180^\circ\).
Both angles are between the parallel lines, on opposite sides of the transversal: a Z shape. Alternate angles are equal, so make an equation.
| \(4x - 15\) | \(=\) | \(2x + 37\) |
| \(2x\) | \(=\) | \(52\) |
| \(x\) | \(=\) | \(\mathbf{26}\) |
Each angle is \(4 \times 26 - 15 = \mathbf{89^\circ}\). Check: \(2 \times 26 + 37 = 89\).
If the angles were co-interior instead, the equation would be \((4x - 15) + (2x + 37) = 180\).
(i) \(x\), the size of \(\angle ABC\)
(ii) \(\angle AED\)
(iii) \(y\), the size of \(\angle BAC\).
(i) \(\angle ADE\) and \(\angle ABC\) sit in matching positions on the parallel lines \(DE\) and \(BC\): an F shape.
| \(x\) | \(=\) | \(\mathbf{58^\circ}\) |
| \(\text{(corresponding, } DE \parallel BC)\) |
(ii) In the same way, using the side \(AC\) as the transversal:
| \(\angle AED\) | \(=\) | \(\angle ACB = \mathbf{67^\circ}\) |
| \(\text{(corresponding, } DE \parallel BC)\) |
(iii) Use the angle sum of \(\triangle ABC\).
| \(y\) | \(=\) | \(180^\circ - 58^\circ - 67^\circ\) |
| \(\) | \(=\) | \(\mathbf{55^\circ}\) |
Check in \(\triangle ADE\): \(58 + 55 + 67 = 180\) \(\checkmark\)
(i) Find \(x = \angle BED\).
(ii) Find the reflex angle \(BED\).
No single transversal joins the two given angles. Draw a third line \(EF\) through \(E\), parallel to \(AB\) and \(CD\). It splits \(x\) into two parts.
(i)
| \(\angle BEF\) | \(=\) | \(42^\circ\ \text{(alternate, } AB \parallel EF)\) |
| \(\angle DEF\) | \(=\) | \(56^\circ\ \text{(alternate, } CD \parallel EF)\) |
| \(x\) | \(=\) | \(42^\circ + 56^\circ = \mathbf{98^\circ}\) |
(ii) The reflex angle goes the long way round \(E\). Angles at a point add to \(360^\circ\).
| \(\text{reflex } \angle BED\) | \(=\) | \(360^\circ - 98^\circ = \mathbf{262^\circ}\) |
If the given angles open the other way (co-interior with \(EF\)), find each part as \(180^\circ\) minus the given angle, then add.
Common pitfalls
Frequently asked questions
What are alternate angles?
Alternate angles are between two parallel lines on opposite sides of the transversal, in a Z shape. They are equal.
What is the difference between co-interior and corresponding angles?
Corresponding angles are in matching positions at each intersection (an F shape) and are equal. Co-interior angles are both between the parallel lines on the same side (a C shape) and add to \(180^\circ\).
How do you show that two lines are parallel?
Find a transversal and check one pair of angles. If corresponding or alternate angles are equal, or co-interior angles add to \(180^\circ\), the lines are parallel.
How do you find an angle at a bend between parallel lines?
Draw a third line through the bend, parallel to the other two. It splits the angle into two parts, and each part is an alternate angle you already know. Add the parts.
What reason should I write for an angle on parallel lines?
Name the angle pair and the parallel lines, for example (alternate angles, \(AB \parallel CD\)) or (co-interior angles, \(AB \parallel CD\)).