Two pairs at every crossing
Two lines cross. Are the angles facing each other equal?
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- Two lines cross at a point.
- Four angles form: a, b, c and d.
- a and b sit on one line, so a + b = 180°.
- b and c sit on the other line, so b + c = 180°.
- Take away the shared b. Then a = c. Here both are 118°.
The rule
Yes. Both angles make a linear pair with the same third angle.
Theorem. Where two lines cross, the vertically opposite angles are equal. These are the angles that face each other. A theorem is a fact we prove.
Name the angles a, b, c and d around the crossing.
A proof is a chain of reasons. The proof uses the linear pair twice. a + b = 180° and b + c = 180°. So a + b = b + c. Take away b from both sides: a + b − b = b + c − b. Then a = c.
Now use b + c = 180° and c + d = 180°. So b + c = c + d. Take away c from both sides. Then b = d.
Example. One angle at a crossing is 118°. The angle facing it is also 118°. Each angle beside it makes a linear pair with it. So each is 180° − 118° = 62°.
The converse (the claim turned around) is false. Two equal angles need not face each other.