Vertically opposite angles | Year 7

01 / LEARN

The idea, step by step

Find the pair across the vertex

When two straight lines cross, they make two pairs of vertically opposite angles. In the diagram, a and c are one pair; b and d are the other. The angles in each pair lie across the vertex and do not share an arm.

Vertically opposite angles are equal. Adjacent angles at this intersection usually are not equal: they add to $180^{\circ}$ . The word vertically refers to the shared vertex, not to whether the picture is upright.

The equality comes from two straight lines

Angle a and angle b total $180^{\circ}$ . Angle c and the same angle b also total $180^{\circ}$ . Subtract b from both totals: a and c must be equal. This is a reason for the rule, not a measurement guess.

If opposite angles are written as expressions, set the expressions equal. Solve for the unknown, then substitute it back to obtain the angle size. Finally check that each adjacent pair still totals $180^{\circ}$ .

A common trap

The two angles next to each other at a crossing are the vertically opposite pair.

Instead: Vertically opposite angles lie across the vertex. Adjacent angles share an arm and, at intersecting straight lines, add to 180 degrees.

02 / WORKED EXAMPLES

See the thinking

Example 1

Complete an intersection

In the diagram, $a = 65^{\circ}$ . Find b, c and d.

c is vertically opposite a, so $c = 65^{\circ}$ .
a and b lie on a straight line, so $b = 180^{\circ} - 65^{\circ} = 115^{\circ}$ .
d is vertically opposite b, so $d = 115^{\circ}$ .

b = 115^{\circ}

c = 65^{\circ}

d = 115^{\circ}

Example 2

Equal expressions

Two vertically opposite angles are ${(3 x + 8)}^{\circ}$ and ${(5 x - 30)}^{\circ}$ . Find x and the angle size.

Vertically opposite angles are equal: $3 x + 8 = 5 x - 30$ .
Add 30 and subtract $3 x$ : $38 = 2 x$ , so $x = 19$ .
Substitute: $3 (19) + 8 = 65$ and $5 (19) - 30 = 65$ .

x = 19

; each marked angle is

65^{\circ}

Check your understanding

One angle at two intersecting straight lines is 71 degrees. What are the sizes of the other three?

Show answer

$71^{\circ}$ , $109^{\circ}$ and $109^{\circ}$ .

The opposite angle is 71 degrees. Each adjacent angle is $180 - 71 = 109$ degrees.

03 / WORKSHEET

Your turn

6 questions · 13 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Two straight lines cross. One angle is $47^{\circ}$ . Find its vertically opposite angle.
1 mark
Use the labelled intersection. Given $a = 130^{\circ}$ , find the following angles.
3 marks
1. (a)
  Find b.
  1 mark
2. (b)
  Find c.
  1 mark
3. (c)
  Find d.
  1 mark
Four equal angles meet where two straight lines cross. Find each angle and name the relationship between the lines.
2 marks
Vertically opposite angles are ${(4 x + 5)}^{\circ}$ and ${(6 x - 25)}^{\circ}$ . Find x and the angle size.
3 marks
Two angles both measure $52^{\circ}$ . Does this fact alone prove they are vertically opposite? Explain.
2 marks
Two straight wires cross. The smaller angle between them is $38^{\circ}$ . Find the larger angle, giving a reason.
2 marks

Read the worked solutions

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

Two straight lines cross. One angle is $47^{\circ}$ . Find its vertically opposite angle.
1 mark
Vertically opposite angles are equal.
$47^{\circ}$ .
Use the labelled intersection. Given $a = 130^{\circ}$ , find the following angles.
3 marks
Use equality for opposite pairs and a 180-degree total for adjacent pairs.
1. (a)
  Find b.
  1 mark
  $180 - 130 = 50$ .
  $50^{\circ}$ .
2. (b)
  Find c.
  1 mark
  c is vertically opposite a.
  $130^{\circ}$ .
3. (c)
  Find d.
  1 mark
  d is vertically opposite b.
  $50^{\circ}$ .
b = 50 degrees; c = 130 degrees; d = 50 degrees.
Four equal angles meet where two straight lines cross. Find each angle and name the relationship between the lines.
2 marks
The four angles total $360^{\circ}$ .
$360 \div 4 = 90$ . Right-angle intersections are perpendicular.
$90^{\circ}$ each; the lines are perpendicular.
Vertically opposite angles are ${(4 x + 5)}^{\circ}$ and ${(6 x - 25)}^{\circ}$ . Find x and the angle size.
3 marks
$4 x + 5 = 6 x - 25$ .
$30 = 2 x$ , so $x = 15$ .
$4 (15) + 5 = 65$ . Check $6 (15) - 25 = 65$ .
$x = 15$ ; the angle is $65^{\circ}$ .
Two angles both measure $52^{\circ}$ . Does this fact alone prove they are vertically opposite? Explain.
2 marks
Vertically opposite describes a position at two intersecting straight lines.
Equal size alone does not supply that arrangement.
No. Equal angles need not be at the same intersection.
Two straight wires cross. The smaller angle between them is $38^{\circ}$ . Find the larger angle, giving a reason.
2 marks
The smaller and larger adjacent angles form a straight angle.
$180^{\circ} - 38^{\circ} = 142^{\circ}$ .
$142^{\circ}$ .

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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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