PROOF 48
Prove.
Vertical angles are equal
On the Euclidean plane.
A way to read this proof
First separate what is assumed from what is to be proved. At each step, ask which definition or earlier result permits the move. If it is unclear, return to that premise rather than skipping to the conclusion.
Definition D1
Two distinct lines ℓ, m meet at a point O. Take two points A, B on ℓ with O between them, and two points C, D on m with O between them. The angles ∠AOC and ∠BOD are called vertical angles.
Proof.
(1)If they met at two points, there would be two lines through those two points; hence ℓ and m meet only at O.A2
(2)Therefore C and D do not lie on line AB, and A and B do not lie on line CD.(1)
(3)∠BOD is the angle obtained from ∠AOC by replacing each side with its opposite ray.convention
∎
Theorem T
Vertical angles are equal. That is, ∠AOC = ∠BOD
Proof.
(1)O is between A and B, so ∠AOB = 180°D1, A1
(2)C does not lie on line AB, so ray OC is interior to ∠AOB.D1, A4
(3)∠AOC + ∠COB = ∠AOB = 180°(1), (2), A3
(4)O is between C and D, so ∠COD = 180°D1, A1
(5)B does not lie on line CD, so ray OB is interior to ∠COD.D1, A4
(6)∠COB + ∠BOD = ∠COD = 180°(4), (5), A3
(7)∠AOC + ∠COB = ∠COB + ∠BOD(3), (6), E
(8)Since ∠COB + ∠BOD = ∠BOD + ∠COB, we have ∠AOC + ∠COB = ∠BOD + ∠COB(7), A6
(9)∠AOC = ∠BOD(8), A5
∎
Remark
Can A1 be removed from this proof?
Distinguish whether the point is the value 180°, or the fact that the two straight angles share one single value.