Parallel lines and transversals | Year 7

01 / LEARN

The idea, step by step

Position first, rule second

A transversal crosses two lines. In the labelled diagram the horizontal lines are parallel. Corresponding positions match at the two intersections: a with e, b with f, c with g, and d with h. These corresponding angles are equal.

Alternate interior angles lie between the parallel lines on opposite sides of the transversal. Here c equals e and d equals f. Mark the two angles before choosing a rule; a rotated diagram has exactly the same relationships.

Some pairs add instead of matching

Co-interior angles lie between the lines on the same side of the transversal. Here c with f, and d with e, each total $180^{\circ}$ . They are not generally equal.

These facts require parallel lines. If parallelism is unknown, the converse is useful: equal corresponding angles, equal alternate interior angles, or co-interior angles totalling $180^{\circ}$ establish that the lines are parallel. The angles must be in the stated positions; two unrelated equal angles are not enough.

c + f = 180^{\circ}

A common trap

All angles between parallel lines are equal.

Instead: Alternate interior angles are equal; co-interior angles add to 180 degrees. Use their positions, not just the fact that both are inside.

02 / WORKED EXAMPLES

See the thinking

Example 1

Use equal and supplementary pairs

The horizontal lines are parallel. If $a = 68^{\circ}$ , find c, e and f.

c is vertically opposite a, so $c = 68^{\circ}$ .
e corresponds to a, so $e = 68^{\circ}$ .
e and f form a straight angle, so $f = 180^{\circ} - 68^{\circ} = 112^{\circ}$ .

c = 68^{\circ}

e = 68^{\circ}

f = 112^{\circ}

Example 2

A co-interior equation

Co-interior angles c and f are ${(3 x + 4)}^{\circ}$ and ${(5 x + 16)}^{\circ}$ . The lines are parallel. Find x.

Co-interior angles total $180^{\circ}$ .
$(3 x + 4) + (5 x + 16) = 180$ , so $8 x + 20 = 180$ .
$8 x = 160$ , hence $x = 20$ . The angles are 64 and 116 degrees, which total 180.

x = 20

Check your understanding

Two co-interior angles on parallel lines are $112^{\circ}$ and x. Find x.

Show answer

$68^{\circ}$ .

Co-interior angles total 180 degrees, so $180 - 112 = 68$ .

03 / WORKSHEET

Your turn

6 questions · 14 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

The horizontal lines in the diagram are parallel and $a = 73^{\circ}$ .
3 marks
1. (a)
  Find e and name the angle rule.
  1 mark
2. (b)
  Find h and give a reason.
  2 marks
The horizontal lines are parallel and $d = 113^{\circ}$ . Find f and name the angle relationship.
2 marks
The horizontal lines are parallel and $c = 65^{\circ}$ . Find f.
2 marks
Corresponding angles b and f are ${(2 x + 12)}^{\circ}$ and ${(4 x - 24)}^{\circ}$ . The lines are parallel. Find x and their common size.
3 marks
Two lines cut by a transversal have co-interior angles of $108^{\circ}$ and $72^{\circ}$ . What can you conclude? Explain.
2 marks
Two corresponding angles are exactly $88^{\circ}$ and $89^{\circ}$ . Are the lines parallel? Explain.
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

The horizontal lines in the diagram are parallel and $a = 73^{\circ}$ .
3 marks
a and e correspond. e and h are adjacent on a straight line.
1. (a)
  Find e and name the angle rule.
  1 mark
  $e = a = 73^{\circ}$ .
  $73^{\circ}$ , corresponding angles.
2. (b)
  Find h and give a reason.
  2 marks
  $h = 180^{\circ} - e = 107^{\circ}$ .
  $107^{\circ}$ .
e = 73 degrees; h = 107 degrees.
The horizontal lines are parallel and $d = 113^{\circ}$ . Find f and name the angle relationship.
2 marks
d and f lie inside the lines on opposite sides of the transversal.
Alternate interior angles are equal.
$f = 113^{\circ}$ , alternate interior angles.
The horizontal lines are parallel and $c = 65^{\circ}$ . Find f.
2 marks
c and f are co-interior.
$f = 180^{\circ} - 65^{\circ} = 115^{\circ}$ .
$f = 115^{\circ}$ .
Corresponding angles b and f are ${(2 x + 12)}^{\circ}$ and ${(4 x - 24)}^{\circ}$ . The lines are parallel. Find x and their common size.
3 marks
$2 x + 12 = 4 x - 24$ .
$36 = 2 x$ , so $x = 18$ .
$2 (18) + 12 = 48$ .
$x = 18$ ; both angles are $48^{\circ}$ .
Two lines cut by a transversal have co-interior angles of $108^{\circ}$ and $72^{\circ}$ . What can you conclude? Explain.
2 marks
$108 + 72 = 180$ .
Co-interior angles totalling 180 degrees establish parallelism.
The lines are parallel.
Two corresponding angles are exactly $88^{\circ}$ and $89^{\circ}$ . Are the lines parallel? Explain.
2 marks
Corresponding angles must be equal when lines are parallel.
$88 \neq 89$ , so the given exact angles do not establish parallel lines; they show the lines are not parallel.
No.

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Curriculum reference & lesson scope

Australian Curriculum v9 / SA Mathematics

Year 7 angle relationships, polygon properties and classification algorithms. Chapter grouping follows the approved Ascent outline.

Official curriculum source ↗

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