Alternate Angles on Parallel Lines
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Theory
Alternate angles lie between two lines, on opposite sides of a transversal. If the lines are parallel, alternate angles are equal. This Year 7 maths page shows how to spot alternate angles (a Z shape) and use them with reasons, including an extra parallel line through a point.
A transversal crossing two lines makes \(8\) angles. The \(4\) angles between the two lines are \(c\), \(d\), \(e\) and \(f\).
Alternate angles are between the lines and on opposite sides of the transversal: \(c\) and \(f\), and \(d\) and \(e\). Together an alternate pair makes a Z shape (which may be back to front).
The rule. Arrow marks show the lines are parallel. Then alternate angles on parallel lines are equal.
With named points (a transversal meets \(AB\) at \(X\) and \(CD\) at \(Y\)), name the parallel lines: “alternate angles, \(AB \parallel CD\)”: \(\angle XYD = \angle AXY\).
Alternate pairs (between the lines, opposite sides):
| Pair | Top crossing | Bottom crossing |
|---|---|---|
| first pair | \(c\) (below, left) | \(f\) (above, right) |
| second pair | \(d\) (below, right) | \(e\) (above, left) |
How to use alternate angles
- Check that the lines are parallel.
- Check both angles are between the lines, on opposite sides of the transversal.
- Write the reason and set them equal.
- Finish with other angle facts if needed.
\(66^\circ\): between the lines, left. \(a^\circ\): between the lines, right. Opposite sides: a Z shape. Alternate angles on parallel lines are equal.
\(a = 66\)
\(a^\circ\) and \(114^\circ\) are alternate. \(b^\circ\) and \(a^\circ\) are on a straight line.
| \(a\) | \(=\) | \(114\) |
| \(b\) | \(=\) | \(180 - 114 = 66\) |
\(a = 114\), \(b = 66\)
| \(\angle RVU\) | \(=\) | \(\angle SVW = 108^\circ\) |
| \(\angle QUV\) | \(=\) | \(\angle RVU = 108^\circ\) |
Vertically opposite angles are equal, then alternate angles, \(PQ \parallel RS\). \(\angle QUV = 108^\circ\).
| \(\angle FGD\) | \(=\) | \(180^\circ - 75^\circ = 105^\circ\) |
| \(\angle AFG\) | \(=\) | \(\angle FGD = 105^\circ\) |
Angles on a straight line add to \(180^\circ\), then alternate angles, \(AB \parallel CD\). \(\angle AFG = 105^\circ\).
| \(\angle BGH\) | \(=\) | \(\angle AGE = 74^\circ\) |
| \(\angle GHC\) | \(=\) | \(\angle BGH = 74^\circ\) |
| \(\angle GHD\) | \(=\) | \(180^\circ - 74^\circ = 106^\circ\) |
Vertically opposite angles are equal; alternate angles, \(AB \parallel CD\); angles on a straight line add to \(180^\circ\).
Draw \(XY\) through \(X\), parallel to \(AB\) and \(CD\) (dashed). It splits \(x^\circ\) into two parts. Alternate angles, \(AB \parallel XY\): the top part is \(38^\circ\). Alternate angles, \(XY \parallel CD\): the bottom part is \(47^\circ\).
| \(x\) | \(=\) | \(38 + 47 = 85\) |
\(x = 85\)
\(x\) and \(\angle ABC\) are alternate (\(PQ \parallel BC\)); so are \(z\) and \(\angle ACB\). Then \(x\), \(y\) and \(z\) make a straight angle at \(A\).
| \(x\) | \(=\) | \(55\) |
| \(z\) | \(=\) | \(65\) |
| \(y\) | \(=\) | \(180 - 55 - 65 = 60\) |
\(x = 55\), \(y = 60\), \(z = 65\)
Common pitfalls
Frequently asked questions
What are alternate angles?
Angles between two lines, on opposite sides of a transversal.
Are alternate angles always equal?
Only when the lines are parallel.
What shape do alternate angles make?
A Z shape, which may be back to front.
How do you find an angle at a point between two parallel lines?
Draw a third parallel line through the point and use two pairs of alternate angles.
What reason do you write?
Alternate angles on parallel lines are equal (or: alternate angles, \(AB \parallel CD\)).