Two lines crossing
When two straight lines cross, they make four angles around the point where they meet. Name them a, b, c and d, going round the point in order.
Two angles that sit next to each other and share an arm, such as a and b, are called adjacent. Two angles that face each other across the crossing, a and c, or b and d, are called vertically opposite. "Vertically" here means "at the vertex", the point the four angles share; it has nothing to do with up and down.
The slanting line crosses the upper line and makes four angles. a and c face each other across the crossing, and so do b and d.
Why opposite angles are equal
Look at a and b. They sit side by side on one of the straight lines, so together they make a half turn: a + b = 180.
Now look at b and c. They sit side by side on the other straight line, so b + c = 180 as well.
Both a and c are what is left when b is taken from 180: a = 180 − b and c = 180 − b. So a = c. The same argument with c in the middle, on the two lines again, gives b = d. Vertically opposite angles are always equal, however the two lines cross.
The two angles facing each other across the crossing are both 70°.
Two equal pairs
So two crossing lines make two pairs of equal angles. If one angle is 70°, the angle opposite it is also 70°. Each neighbor of the 70° angle is 180 − 70 = 110°, and those two neighbors are opposite each other, so both are 110°.
The four angles are 70°, 110°, 70° and 110°. Check: they go all the way round the point, and 70 + 110 + 70 + 110 = 360.
Knowing one angle at a crossing gives all four: the opposite angle is the same, and the two neighbors are 180 minus it. The only crossing where all four angles are equal is a square one, where each is 90°.
Two equal pairs: 70° twice and 110° twice. Each 70° angle and a 110° angle beside it make 180°.
The usual mistakes
Taking the neighbor for the opposite angle. If one angle is 38°, the angle opposite it is 38°, but the angle next to it is 180 − 38 = 142°. The neighbor and the angle add to 180; the opposite angle is a copy.
Using a full turn. The opposite angle is not 360 − 38 = 322°. That would be the whole way round the point except for the 38° angle.
Using the rule where there are no straight lines. The two angles must be made by the same two straight lines crossing. Two rays that meet at a point, or a line that bends at the crossing, do not make vertically opposite angles.
More than two lines
Three straight lines through one point make six angles around it. Each angle is still vertically opposite another angle, the one made by the same two lines on the far side of the point, and the two are equal.
An angle is named with three letters, the vertex in the middle: ∠AOC is the angle at O between OA and OC. The angles on one side of a straight line through O still add up to 180°, however many rays split them.
Worked example: Vertically Opposite Angles at a Three-Line Crossing
Question Three straight lines AB, CD and EF meet at O. ∠ AOC = 38° and ∠ COE = 65°. Find ∠ BOD and ∠ AOF.
1.AB, CD and EF are straight lines through O, so each pair of opposite angles at O is equal.
Three straight lines through O. Two angles on the upper side of AB are given. 2.∠ BOD is vertically opposite ∠ AOC: ∠ BOD = 38°.
∠ BOD is across O from ∠ AOC, on the same two lines: 38°. 3.Along the straight line AB, the angles on the upper side add to 180°: ∠ AOC + ∠ COE + ∠ EOB = 180°.
Along the upper side of the straight line AB the three angles add to 180°. 4.∠ EOB = 180° − 38° − 65° = 77°.
∠ EOB = 180° − 38° − 65° = 77°. 5.∠ AOF is vertically opposite ∠ EOB: ∠ AOF = 77°.
∠ AOF is across O from ∠ EOB: 77°.
Answer: ∠ BOD = 38°; ∠ AOF = 77°
Common mistakes
- Pairing ∠ BOD with ∠ COE because both sit near D: opposite angles share the vertex and lie across it, on the same two lines.
- Taking ∠ AOF = 65°: OF is opposite OE, so ∠ AOF matches the angle from OE to OB, not the one from OC to OE.