Angles on Parallel Lines
Every question with a fully worked solution.
- Angles on Parallel Lines - Video - Parallel lines and transversals | Find the Angle Measure Watch
Theory
When a transversal crosses parallel lines, corresponding, alternate and co-interior angles let you find every unknown angle, one step and one reason at a time. This Year 7 maths page compares the three pairs and works through problems that combine them with other angle facts.
A transversal across two parallel lines makes eight angles, but only two sizes. All the acute angles are equal, all the obtuse angles are equal, and an acute angle and an obtuse angle add to \(180^\circ\).
Corresponding angles (F shape) and alternate angles (Z shape) are equal. Co-interior angles (C shape) are supplementary: they add to \(180^\circ\).
You can also use: angles on a straight line add to \(180^\circ\); vertically opposite angles are equal; angles at a point add to \(360^\circ\).
Two sets of parallel lines? Single arrow marks show one pair, double arrow marks show the other. In a parallelogram each side is a transversal for the other pair, so angles next to each other are co-interior.
The three pairs:
| Pair | Shape | Rule |
|---|---|---|
| corresponding | F | equal |
| alternate | Z | equal |
| co-interior | C | add to \(180^\circ\) |
How to find an angle on parallel lines
- Find a pair that links a known angle to the unknown.
- Write its reason, with the parallel lines named.
- Write one equation and solve it.
- Repeat until you reach the angle you want, then check it against the picture.
(i) co-interior to \(\angle AFG\)?
(ii) alternate to \(\angle AFG\)?
\(\angle AFG\) is between the lines, left of the transversal. Same side at \(G\): \(\angle CGF\). Opposite side at \(G\): \(\angle DGF\).
(i) \(\angle CGF\)
(ii) \(\angle DGF\)
(i) \(\angle QUT\)?
(ii) \(\angle PUV\)?
\(\angle QUT\) is above \(PQ\), right of the transversal; at \(V\) that position is \(\angle SVU\). \(\angle PUV\) is below \(PQ\), left; at \(V\) that is \(\angle RVW\).
(i) \(\angle SVU\)
(ii) \(\angle RVW\)
\(x^\circ\) and \(53^\circ\): between the lines, opposite sides, so alternate angles, \(AB \parallel CD\). \(y^\circ\) and \(53^\circ\): between the lines, same side, so co-interior angles, \(AB \parallel CD\).
| \(x\) | \(=\) | \(53\) |
| \(y + 53\) | \(=\) | \(180\) |
| \(y\) | \(=\) | \(127\) |
\(x = 53\), \(y = 127\)
| \(\angle GHD\) | \(=\) | \(\angle AGH = 126^\circ\) |
| \(\angle DHF\) | \(=\) | \(180^\circ - 126^\circ = 54^\circ\) |
Alternate angles, \(AB \parallel CD\); then angles on a straight line add to \(180^\circ\).
Both angles are between the lines, on the right: co-interior angles, \(AB \parallel CD\).
| \((2x + 10) + 100\) | \(=\) | \(180\) |
| \(2x\) | \(=\) | \(70\) |
| \(x\) | \(=\) | \(35\) |
\(x = 35\). Check: \(2 \times 35 + 10 = 80\) and \(80 + 100 = 180\).
Draw \(XY\) through \(X\), parallel to \(AB\) and \(CD\), pointing away from \(A\) and \(C\). Co-interior angles, \(AB \parallel XY\): \(\angle AXY = 180 - 36 = 144^\circ\). Co-interior angles, \(CD \parallel XY\): \(\angle CXY = 180 - 47 = 133^\circ\). Angles at a point add to \(360^\circ\):
| \(x + 144 + 133\) | \(=\) | \(360\) |
| \(x\) | \(=\) | \(83\) |
\(x = 83\). (Alternate angles give the same answer: \(36 + 47 = 83\).)
Each side is a transversal for the other pair of parallel sides, so angles next to each other are co-interior.
| \(\angle PQR\) | \(=\) | \(180^\circ - 74^\circ = 106^\circ\) |
| \(\angle QRS\) | \(=\) | \(180^\circ - 106^\circ = 74^\circ\) |
| \(\angle RSP\) | \(=\) | \(180^\circ - 74^\circ = 106^\circ\) |
\(106^\circ\), \(74^\circ\) and \(106^\circ\). Opposite angles of a parallelogram are equal.
Common pitfalls
Frequently asked questions
Which angles on parallel lines are equal?
Corresponding angles and alternate angles.
Which angles on parallel lines add to 180°?
Co-interior angles (and angles on a straight line).
How many different sizes are the eight angles?
Two (or one, if the transversal is perpendicular): every acute angle is equal and every obtuse angle is equal.
How do you find an angle at a point between parallel lines?
Draw a third parallel line through the point, then use alternate or co-interior angles.
What are the angles of a parallelogram?
Angles next to each other add to \(180^\circ\); opposite angles are equal.