Vertically opposite angles | Year 6

01 / LEARN

Understand the idea.

Opposite at an intersection

When two straight lines cross, the angles directly across the vertex from one another are vertically opposite angles. Each opposite pair is equal. The word vertically names the relationship; the diagram need not point up and down.

Opposite angles share the crossing point but no arm. Neighbouring, or adjacent, angles share an arm and form a straight angle. Identify which relationship connects the labels before deciding whether to copy a value or subtract from $180^{\circ}$ .

Link equality to a straight line

The straight-line rule explains opposite-angle equality. If an angle is $65^{\circ}$ , its neighbour is $180^{\circ} - 65^{\circ} = 115^{\circ}$ . The next angle is $180^{\circ} - 115^{\circ} = 65^{\circ}$ , so it matches the angle opposite it.

The same reasoning works for any crossing of two straight lines: both opposite angles are what remains when the same neighbouring angle is removed from $180^{\circ}$ . As a check, all four angles around the crossing total $360^{\circ}$ .

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

The opposite angle

Two lines intersect as shown. Find x opposite the 65° angle.

Worked solution

1
Identify the opposite pair.
$x is vertically opposite 65^{\circ}$
The two angles face each other across the crossing and do not share an arm.
2
Use opposite-angle equality.
$x = 65^{\circ}$
Vertically opposite angles formed by two straight lines are equal.

Answer

65°

EXAMPLE 02

All four angles

At a crossing, one angle is 110°. Find the angle opposite it and the two neighbouring angles.

Worked solution

1
Find the directly opposite angle.
$x = 110^{\circ}$
Vertically opposite angles are equal.
2
Find a neighbouring angle.
$y = 180^{\circ} - 110^{\circ} = 70^{\circ}$
This neighbour and the given angle form a straight angle.
3
Find the other neighbouring angle.
$other neighbour = 70^{\circ}$
The two neighbouring angles are themselves a vertically opposite pair.
4
Check the complete crossing.
$110^{\circ} + 70^{\circ} + 110^{\circ} + 70^{\circ} = 360^{\circ}$
All four angles make one complete turn around the intersection.

Answer

Opposite: 110°; each neighbour: 70°.

03 / YOUR TURN

Try it yourself.

Check your understanding

Two lines cross. An angle is 83°. Find the angle directly opposite it.

Show answer & reasoning

83°

The requested angle is directly opposite the $83^{\circ}$ angle. Vertically opposite angles are equal, so it is also $83^{\circ}$ .

Keep practising

6 worksheet questions · 11 marks

Open worksheet Worked solutions

Read the questions online

Find $x$ in the crossing shown.
1 mark
Two lines cross. One angle is $105^{\circ}$ . Find an adjacent angle.
1 mark
At a crossing one angle is $38^{\circ}$ . List all four angles in order around the point.
2 marks
One angle at a crossing is $90^{\circ}$ . What are the other three? Explain.
2 marks
A crossing has adjacent angles marked $112^{\circ}$ and $112^{\circ}$ . Could both marks be correct? Explain.
2 marks
At an intersection, $x$ is opposite $127^{\circ}$ and $y$ is next to $127^{\circ}$ . Find $x$ and $y$ , giving a reason for each.
3 marks

Read the worked solutions

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

Find $x$ in the crossing shown.
1 mark
$x$ is vertically opposite the $54^{\circ}$ angle, so $x = 54^{\circ}$ .
$54^{\circ}$
Two lines cross. One angle is $105^{\circ}$ . Find an adjacent angle.
1 mark
$180^{\circ} - 105^{\circ} = 75^{\circ}$ .
$75^{\circ}$
At a crossing one angle is $38^{\circ}$ . List all four angles in order around the point.
2 marks
An adjacent angle is $180^{\circ} - 38^{\circ} = 142^{\circ}$ .
Opposite pairs are equal.
$38^{\circ}$ , $142^{\circ}$ , $38^{\circ}$ , $142^{\circ}$ (starting at the given angle)
One angle at a crossing is $90^{\circ}$ . What are the other three? Explain.
2 marks
The opposite angle equals $90^{\circ}$ .
Each neighbour is $180^{\circ} - 90^{\circ} = 90^{\circ}$ .
All are $90^{\circ}$ .
A crossing has adjacent angles marked $112^{\circ}$ and $112^{\circ}$ . Could both marks be correct? Explain.
2 marks
Adjacent angles on the two straight lines must total $180^{\circ}$ .
$112^{\circ} + 112^{\circ} = 224^{\circ}$ , not $180^{\circ}$ .
No.
At an intersection, $x$ is opposite $127^{\circ}$ and $y$ is next to $127^{\circ}$ . Find $x$ and $y$ , giving a reason for each.
3 marks
$x$ is vertically opposite $127^{\circ}$ , so they are equal.
$y$ forms a straight angle with $127^{\circ}$ , so $y = 180^{\circ} - 127^{\circ} = 53^{\circ}$ .
$x = 127^{\circ}$ ; $y = 53^{\circ}$

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Curriculum reference & lesson scope

Australian Curriculum v9 / SA Curriculum Mathematics

Year 6 angle relationships: straight-line angles, angles around a point, vertically opposite angles and connected reasoning.

Official curriculum source ↗

In this lesson

Understand the idea.

Opposite at an intersection

Link equality to a straight line

See the method in action.

The opposite angle

Identify the opposite pair.

Use opposite-angle equality.

All four angles

Find the directly opposite angle.

Find a neighbouring angle.

Find the other neighbouring angle.

Check the complete crossing.

Try it yourself.

Check your understanding

Keep practising