Parallel lines and transversals | Year 7

01 / LEARN

Understand the idea.

Position first, rule second

A transversal crosses two lines. When the lines are parallel, corresponding angles in matching positions are equal: $a = e$ , $b = f$ , $c = g$ and $d = h$ in the supplied diagram. Match the positions before choosing the rule.

Alternate interior angles are inside the parallel lines on opposite sides of the transversal. Here $c = e$ and $d = f$ . Rotating the whole diagram preserves these relationships, so use the angle positions rather than an upright letter shape.

Some pairs add instead of matching

Co-interior angles are inside the lines on the same side of the transversal. With parallel lines their sum is $180^{\circ}$ : here $c + f = 180^{\circ}$ and $d + e = 180^{\circ}$ . Being inside does not mean the angles are equal.

These conclusions need parallel lines. Conversely, if a transversal forms equal corresponding angles, equal alternate interior angles, or co-interior angles totalling $180^{\circ}$ , that relationship establishes parallelism. Equal angles in unrelated positions are insufficient.

c + f = 180^{\circ}

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Use equal and supplementary pairs

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

Worked solution

1
Use the opposite angle at the upper intersection.
$c = a = 68^{\circ}$
Angles a and c are vertically opposite.
2
Match the corresponding positions.
$e = a = 68^{\circ}$
The lines are parallel and a and e occupy corresponding positions.
3
Use the straight angle at the lower intersection.
$f = 180^{\circ} - 68^{\circ} = 112^{\circ}$
Angles e and f are adjacent on the lower straight line.
4
Check the co-interior pair.
$c + f = 68^{\circ} + 112^{\circ} = 180^{\circ}$
The calculated interior angles on the same side of the transversal have the required total.

Answer

$c = 68^{\circ}$ , $e = 68^{\circ}$ , $f = 112^{\circ}$ .

EXAMPLE 02

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.

Worked solution

1
Set up the co-interior total.
$(3 x + 4) + (5 x + 16) = 180$
Co-interior angles between parallel lines total 180 degrees.
2
Combine like terms.
$8 x + 20 = 180$
The variable terms total $8 x$ and the constants total 20.
3
Subtract 20, then divide by 8.
$\begin{aligned} 8 x & = 160 \\ x & = 20 \end{aligned}$
The same inverse operations on both sides isolate x.
4
Check the two angle sizes.
$\begin{aligned} 3 (20) + 4 & = 64 \\ 5 (20) + 16 & = 116 \\ 64^{\circ} + 116^{\circ} & = 180^{\circ} \end{aligned}$
The substituted angles satisfy the required supplementary relationship.

Answer

$x = 20$ .

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

$68^{\circ}$ .

Co-interior angles on parallel lines total $180^{\circ}$ . Therefore $x = 180^{\circ} - 112^{\circ} = 68^{\circ}$ , and $112^{\circ} + 68^{\circ} = 180^{\circ}$ .

Keep practising

6 worksheet questions · 14 marks

Open worksheet 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 ↗

In this lesson

Understand the idea.

Position first, rule second

Some pairs add instead of matching

See the method in action.

Use equal and supplementary pairs

Use the opposite angle at the upper intersection.

Match the corresponding positions.

Use the straight angle at the lower intersection.

Check the co-interior pair.

A co-interior equation

Set up the co-interior total.

Combine like terms.

Subtract 20, then divide by 8.

Check the two angle sizes.

Try it yourself.

Check your understanding

Keep practising