The idea, step by step
Opposite at an intersection
When two straight lines cross, they form four angles. The angles directly opposite each other are equal. These pairs are called vertically opposite angles, even if the diagram is turned sideways.
Opposite angles share only the vertex. Adjacent angles share an arm. Do not call neighbouring angles vertically opposite just because one is above another on the page.
Link equality to a straight line
You can explain why opposite angles are equal using the 180° straight-line rule. If one angle is 65°, each neighbouring angle is 180° − 65° = 115°. The angle opposite the 65° angle is then 180° − 115° = 65°.
A reliable method is to identify the relationship before calculating. Use equality for an opposite angle and subtraction from 180° for an adjacent angle.
A common trap
Setting an adjacent angle equal to the given angle in every crossing.
Instead: Only opposite pairs must be equal. Adjacent angles add to 180°; they are equal only in a right-angle crossing.
See the thinking
The opposite angle
Two lines intersect as shown. Find x opposite the 65° angle.
- x and 65° are vertically opposite.
- Vertically opposite angles are equal.
All four angles
At a crossing, one angle is 110°. Find the angle opposite it and the two neighbouring angles.
- The opposite angle is also 110°.
- Each neighbour forms a straight angle with 110°, so each is 70°.
- Check: 110 + 70 + 110 + 70 = 360.
Check your understanding
Two lines cross. An angle is 83°. Find the angle directly opposite it.
Show answer
83°
They are vertically opposite angles.
Your turn
6 questions · 11 marks · Work at your own pace.
Read the questions online
- Find x in the crossing shown.1 mark
- Two lines cross. One angle is 105°. Find an adjacent angle.1 mark
- At a crossing one angle is 38°. List all four angles in order around the point.2 marks
- One angle at a crossing is 90°. What are the other three? Explain.2 marks
- A crossing has adjacent angles marked 112° and 112°. Could both marks be correct? Explain.2 marks
- At an intersection, x is opposite 127° and y is next to 127°. 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° angle, so x = 54°.
54° - Two lines cross. One angle is 105°. Find an adjacent angle.1 mark
180° − 105° = 75°.
75° - At a crossing one angle is 38°. List all four angles in order around the point.2 marks
An adjacent angle is 180° − 38° = 142°.
Opposite pairs are equal.
38°, 142°, 38°, 142° (starting at the given angle) - One angle at a crossing is 90°. What are the other three? Explain.2 marks
The opposite angle equals 90°.
Each neighbour is 180° − 90° = 90°.
All are 90°. - A crossing has adjacent angles marked 112° and 112°. Could both marks be correct? Explain.2 marks
Adjacent angles on the two straight lines must total 180°.
112° + 112° = 224°, not 180°.
No. - At an intersection, x is opposite 127° and y is next to 127°. Find x and y, giving a reason for each.3 marks
x is vertically opposite 127°, so they are equal.
y forms a straight angle with 127°, so y = 180° − 127° = 53°.
x = 127°; y = 53°
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 ↗