Vertically opposite angles (intersecting lines)
Theory
When two straight lines cross, the angles opposite each other across the crossing point are equal. No calculation is needed — an angle equals the one across from it.
Two crossing straight lines make four angles that share the same point, called the vertex.
The two angles that face each other across the vertex are vertically opposite, and they are always equal.
The crossing makes two such pairs. The two angles that sit side by side on a line are not equal — they add to \(180^\circ\).
Two crossing lines make two equal pairs.
| Pair | Relationship |
|---|---|
| Opposite angles | equal |
| Angles side by side | add to \(180^\circ\) |
How to use vertically opposite angles
- Find the vertex where the two straight lines cross.
- Look straight across from the known angle to the one opposite it.
- Write the opposite angle as equal; for a neighbour on the line, subtract from \(180^\circ\).
Vertically opposite angles are equal.
The opposite angle is \(60^\circ\).
The rule works for obtuse angles too.
The opposite angle is \(120^\circ\).
\(180^\circ - 70^\circ = 110^\circ\).
The neighbour on the line is \(110^\circ\).
Each pair faces across the vertex: \(a = c\) and \(b = d\).
The two opposite pairs are equal.
Common pitfalls
Frequently asked questions
What are vertically opposite angles?
They are the pair of angles facing each other across the point where two straight lines cross. They are always equal.
Are vertically opposite angles always equal?
Yes, whenever two straight lines cross. It does not matter whether the angles are acute or obtuse.
How is a neighbour angle different?
An angle next to it on the line is not opposite. Those two sit on a straight line, so they add to \(180^\circ\).
Why are they equal?
Each of the two angles shares a straight line with the same third angle, so both must be \(180^\circ\) minus that angle, which makes them equal.