Corresponding Angles on Parallel Lines
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Theory
When a transversal crosses two lines, corresponding angles are in the same position at each crossing. If the lines are parallel, corresponding angles are equal. This Year 7 maths page shows how to spot corresponding angles (an F shape) and use them, with reasons, to find unknown angles.
A transversal crosses two lines and makes \(8\) angles, \(4\) at each crossing. Describe each angle by its position: above or below the line, and left or right of the transversal.
Corresponding angles are in the same position at each crossing. In the diagram the pairs are \(a\) and \(e\), \(b\) and \(f\), \(c\) and \(g\), \(d\) and \(h\). Together a corresponding pair makes an F shape (the F may be flipped or upside down).
The rule. Arrow marks show the lines are parallel. Then corresponding angles on parallel lines are equal.
With named points, name the parallel lines in the reason: “corresponding angles, \(AB \parallel CD\)”.
Corresponding pairs (same position):
| Position | Top crossing | Bottom crossing |
|---|---|---|
| above, left | \(a\) | \(e\) |
| above, right | \(b\) | \(f\) |
| below, left | \(c\) | \(g\) |
| below, right | \(d\) | \(h\) |
Corresponding angles on parallel lines are equal.
How to use corresponding angles
- Check that the lines are parallel.
- Describe the position of the known angle and the unknown angle.
- If the positions match, write the reason and set them equal.
- Finish with other angle facts if needed.
\(112^\circ\) is below the top line, left. \(p^\circ\) is below the bottom line, left. Same position: corresponding angles on parallel lines are equal.
\(p = 112\)
\(a^\circ\) and \(74^\circ\) are corresponding. \(b^\circ\) and \(a^\circ\) are on a straight line.
| \(a\) | \(=\) | \(74\) |
| \(b\) | \(=\) | \(180 - 74 = 106\) |
\(a = 74\), \(b = 106\)
| \(\angle QUV\) | \(=\) | \(\angle PUT = 58^\circ\) |
| \(\angle SVW\) | \(=\) | \(\angle QUV = 58^\circ\) |
\(\angle QUV = \angle PUT\) (vertically opposite angles are equal). \(\angle SVW = \angle QUV\) (corresponding angles, \(PQ \parallel RS\)). So \(\angle SVW = 58^\circ\).
| \(\angle BFE\) | \(=\) | \(180^\circ - 102^\circ = 78^\circ\) |
| \(\angle DGF\) | \(=\) | \(\angle BFE = 78^\circ\) |
Angles on a straight line add to \(180^\circ\), then corresponding angles, \(AB \parallel CD\). \(\angle DGF = 78^\circ\).
| \(\angle AGH\) | \(=\) | \(\angle CHF = 66^\circ\) |
| \(\angle BGH\) | \(=\) | \(180^\circ - 66^\circ = 114^\circ\) |
\(\angle AGH\) and \(\angle CHF\) are both below their line, left of \(EF\): corresponding angles, \(AB \parallel CD\). \(\angle BGH\) is on a straight line with \(\angle AGH\).
Corresponding angles: the angle above-left at the bottom crossing is \(117^\circ\). Vertically opposite angles are equal: \(2x + 17 = 117\). Angles on a straight line: \(y = 180 - 117\).
| \(2x + 17\) | \(=\) | \(117\) |
| \(x\) | \(=\) | \(50\) |
| \(y\) | \(=\) | \(180 - 117 = 63\) |
\(x = 50\), \(y = 63\)
Common pitfalls
Frequently asked questions
What are corresponding angles?
Angles in the same position at each crossing of a transversal.
Are corresponding angles always equal?
Only when the lines are parallel.
What shape do corresponding angles make?
An F shape, which may be flipped or upside down.
How many angles does a transversal make with two lines?
\(8\): \(4\) at each crossing.
What reason do you write?
Corresponding angles on parallel lines are equal (or: corresponding angles, \(AB \parallel CD\)).