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Year 7 Maths Angles

Corresponding Angles on Parallel Lines

20 practice questions 1 video lesson Theory + worked examples

When a transversal crosses two lines, corresponding angles sit in the same position at each crossing. In NSW Year 7 maths (NESA outcome MA4‑ANG‑C‑01) you learn that these angles are equal when the two lines are parallel.

You will spot the F shape, even when it is flipped or upside down, find unknown angles and write the reason with the parallel lines named. You will also explain why the rule cannot be used when no arrow marks show that the lines are parallel. Lane markings crossed by a pedestrian path show the same pattern. This subtopic is part of Year 7 Angles.

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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, \(AB \parallel CD\)”.

Eight angles made by a transversal Two parallel lines AB and CD cut by a transversal. The four angles at the top crossing are a (above left), b (above right), c (below left), d (below right); at the bottom crossing e, f, g, h in the same positions. b and f are red. a b c d e f g h A B C D
\(b\) is above \(AB\) and right of the transversal; \(f\) is in the same position at \(CD\). They form an F shape, so \(b = f\).

Corresponding pairs (same position):

PositionTop crossingBottom crossing
above, left\(a\)\(e\)
above, right\(b\)\(f\)
below, left\(c\)\(g\)
below, right\(d\)\(h\)

Corresponding angles on parallel lines are equal.

Check for arrow marks first. Corresponding angles are equal only when the lines are parallel. With no arrow marks and no \(\parallel\) statement, you cannot use the rule.
Use the other facts too. Vertically opposite angles are equal, and angles on a straight line add to \(180^\circ\).

How to use corresponding angles

  1. Check that the lines are parallel.
  2. Describe the position of the known angle and the unknown angle.
  3. If the positions match, write the reason and set them equal.
  4. Finish with other angle facts if needed.
Example 1 — Find a corresponding angle
Find the value of \(p\).
Corresponding angles 112 and p Parallel lines cut by a transversal. 112 degrees is below the top line, left of the transversal; p is below the bottom line, left. 112° p°
Solution

\(112^\circ\) is below the top line, left. \(p^\circ\) is below the bottom line, left. Same position: corresponding angles on parallel lines are equal.

\(p = 112\)

Example 2 — Two steps
Find the values of \(a\) and \(b\).
Angles 74, a and b Parallel lines. 74 degrees above-right at the top crossing; a above-right and b above-left at the bottom crossing. 74° a° b°
Solution

\(a^\circ\) and \(74^\circ\) are corresponding. \(b^\circ\) and \(a^\circ\) are on a straight line.

\(a\)\(=\)\(74\)
\(b\)\(=\)\(180 - 74 = 106\)
a =74, b =180−74=106

\(a = 74\), \(b = 106\)

Example 3 — Vertically opposite, then corresponding
\(PQ \parallel RS\) and \(\angle PUT = 58^\circ\). Find the size of \(\angle SVW\).
Angle PUT is 58 degrees Parallel lines PQ and RS cut by transversal TW at U and V. Angle PUT, above PQ on the left, is 58 degrees. 58° P Q R S T W U V
Solution
\(\angle QUV\)\(=\)\(\angle PUT = 58^\circ\)
\(\angle SVW\)\(=\)\(\angle QUV = 58^\circ\)
QUV = PUT =58, SVW = QUV =58

\(\angle QUV = \angle PUT\) (vertically opposite angles are equal). \(\angle SVW = \angle QUV\) (corresponding angles, \(PQ \parallel RS\)). So \(\angle SVW = 58^\circ\).

Example 4 — Straight line, then corresponding
\(AB \parallel CD\) and \(\angle BFG = 102^\circ\). Find the size of \(\angle DGF\).
Angle BFG is 102 degrees Parallel lines AB and CD cut by transversal EH at F and G. Angle BFG, below AB on the right, is 102 degrees. 102° A B C D E H F G
Solution
\(\angle BFE\)\(=\)\(180^\circ - 102^\circ = 78^\circ\)
\(\angle DGF\)\(=\)\(\angle BFE = 78^\circ\)
BFE =180−102=78, DGF = BFE =78

Angles on a straight line add to \(180^\circ\), then corresponding angles, \(AB \parallel CD\). \(\angle DGF = 78^\circ\).

Example 5 — Naming the angles
\(AB \parallel CD\) and \(\angle CHF = 66^\circ\). Find the sizes of \(\angle AGH\) and \(\angle BGH\).
Angle CHF is 66 degrees Parallel lines AB and CD cut by transversal EF at G and H. Angle CHF, below CD on the left, is 66 degrees. 66° A B C D E F G H
Solution
\(\angle AGH\)\(=\)\(\angle CHF = 66^\circ\)
\(\angle BGH\)\(=\)\(180^\circ - 66^\circ = 114^\circ\)
AGH = CHF =66, BGH =180−66=114

\(\angle AGH\) and \(\angle CHF\) are both below their line, left of \(EF\): corresponding angles, \(AB \parallel CD\). \(\angle BGH\) is on a straight line with \(\angle AGH\).

Example 6 — An equation from the angles
Parallel lines are cut by a transversal. At the top crossing, the angle above the line on the left is \(117^\circ\). At the bottom crossing, the angle below the line on the right is \((2x + 17)^\circ\), and the angle above the line on the right is \(y^\circ\). Find \(x\) and \(y\).
Angles 117, 2x + 17 and y Parallel lines. 117 degrees above-left at the top crossing; (2x + 17) degrees below-right and y degrees above-right at the bottom crossing. 117° (2x+17)° y°
Solution

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: \(y = 180 - 117\).

\(2x + 17\)\(=\)\(117\)
\(x\)\(=\)\(50\)
\(y\)\(=\)\(180 - 117 = 63\)
2 x +17=117, x =50, y =63

\(x = 50\), \(y = 63\)

Common pitfalls

No arrow marks. Without parallel lines, corresponding angles do not have to be equal.
Matching only half the position. \(b\) is above the line and right of the transversal. Its partner must be both; \(h\) is right but below.
Adding to 180 instead of being equal. Corresponding angles are equal. Angles that add to \(180^\circ\) are on a straight line or are co-interior.
Leaving out the reason. Write the reason, with the parallel lines named, before each equation.

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, \(AB \parallel CD\)).

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