Vertically opposite angles | Year 7

01 / LEARN

Understand the idea.

Find the pair across the vertex

Two intersecting straight lines create two pairs of vertically opposite angles. Each pair lies across the vertex without a shared arm: in this diagram, $a$ pairs with $c$ , and $b$ pairs with $d$ . The two angles in each pair are equal.

Adjacent angles at the crossing share an arm and total $180^{\circ}$ . They are equal only in the special case of perpendicular lines. Rotating the drawing changes neither relationship; “vertically opposite” refers to the shared vertex, not the page direction.

The equality comes from two straight lines

The equality follows from straight angles. Both $a + b$ and $c + b$ equal $180^{\circ}$ . Subtracting the same $b$ from those totals gives $a = c$ . This explains the rule without assuming that a sketch is accurately measured.

For opposite angles labelled with expressions, equate their degree values and solve. Then substitute into both expressions to check they agree. Keep the variable value separate from the angle size: they need not be the same number.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Complete an intersection

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

Worked solution

1
Find the angle opposite a.
$c = a = 65^{\circ}$
Vertically opposite angles at intersecting straight lines are equal.
2
Use the adjacent straight angle.
$b = 180^{\circ} - 65^{\circ} = 115^{\circ}$
Angles a and b fill a straight angle.
3
Find the angle opposite b.
$d = b = 115^{\circ}$
The second vertically opposite pair has equal angles.
4
Check the full turn.
$65^{\circ} + 115^{\circ} + 65^{\circ} + 115^{\circ} = 360^{\circ}$
All four calculated regions fill the turn around the intersection.

Answer

$b = 115^{\circ}$ , $c = 65^{\circ}$ , $d = 115^{\circ}$ .

EXAMPLE 02

Equal expressions

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

Worked solution

1
Equate the opposite angle values.
$3 x + 8 = 5 x - 30$
Vertically opposite angles are equal, so their degree measures agree.
2
Subtract $3 x$ from both sides.
$8 = 2 x - 30$
Removing the same variable term from both sides preserves equality.
3
Add 30, then divide by 2.
$\begin{aligned} 38 & = 2 x \\ x & = 19 \end{aligned}$
Inverse operations isolate the variable.
4
Calculate both angle measures.
$\begin{aligned} 3 (19) + 8 & = 65 \\ 5 (19) - 30 & = 65 \end{aligned}$
Both expressions give 65 degrees, confirming $x = 19$ and the common angle size.

Answer

$x = 19$ ; each marked angle is $65^{\circ}$ .

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

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

The angle opposite the given one is also $71^{\circ}$ . Each adjacent angle is $180^{\circ} - 71^{\circ} = 109^{\circ}$ , so the other three are $71^{\circ}$ , $109^{\circ}$ and $109^{\circ}$ .

Keep practising

6 worksheet questions · 13 marks

Open worksheet 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 ↗

In this lesson

Understand the idea.

Find the pair across the vertex

The equality comes from two straight lines

See the method in action.

Complete an intersection

Find the angle opposite a.

Use the adjacent straight angle.

Find the angle opposite b.

Check the full turn.

Equal expressions

Equate the opposite angle values.

Subtract 3⁢x from both sides.

Add 30, then divide by 2.

Calculate both angle measures.

Try it yourself.

Check your understanding

Keep practising

Subtract $3 x$ from both sides.