Definition D1
Triangle ABC consists of three non-collinear points A, B, C and the three segments AB, BC, CA. Its interior angles are ∠BAC, ∠ABC, ∠BCA.
Proof.
(1)The three points do not lie on one line.convention
(2)When a line t cuts two lines ℓ, m at P and Q respectively, the two angles lying in opposite half-planes of t are called alternate angles.convention
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Lemma L1
If a line t cuts two parallel lines ℓ, m, the alternate angles are equal.
Proof.
(1)Suppose t cuts ℓ at P and m at Q.assumption
(2)Since ℓ ∥ m and ℓ ≠ m, Q does not lie on ℓ.assumption
(3)Let α be the alternate angle at P and β the alternate angle at Q.D1
(4)There is a unique line m′ through Q whose alternate angle with t is α.(1), (3), A4
(5)The alternate angles are equal, so m′ ∥ ℓ(4), A3
(6)There is only one line through Q parallel to ℓ, so m′ = m(2), (5), assumption, A2
(7)The alternate angle at m is the alternate angle at m′, so β = α(4), (6), A4
∎
Theorem T
In triangle ABC, ∠BAC + ∠ABC + ∠BCA = 180°
Proof.
(1)A does not lie on line BC, so there is a unique line ℓ through A parallel to line BC.D1, A2
(2)Take two points on ℓ with A between them.(1), A10
(3)Segment BC does not meet ℓ, so B and C lie on the same side of ℓ. In particular B and C are not on ℓ.(1), A6
(4)Line AB passes through B, so it differs from ℓ, and the two lines meet only at A.(1), (3)
(5)The segment joining the two points of (2) meets line AB at A, so the two points lie on opposite sides of line AB.(2), (4), A6
(6)C is not on line AB, so exactly one of the two points is on the same side as C. Call it E and the other D.D1, (5), A6
(7)B is not on ℓ, so ray AB is interior to the straight angle ∠DAE.(3), (6), A8
(8)∠DAB + ∠BAE = ∠DAE = 180°(7), A5, A1
(9)C is on the E side of line AB and on the B side of line AE = ℓ, so ray AC is interior to ∠BAE.(3), (6), A7
(10)∠BAC + ∠CAE = ∠BAE(9), A5
(11)∠DAB + ∠BAC + ∠CAE = 180°(8), (10), E
(12)Line AB cuts ℓ and line BC, and D is on the opposite side to C, so ∠DAB and ∠ABC are alternate angles.(1), (6), D1
(13)∠DAB = ∠ABC(12), L1
(14)By (9), B and E lie on opposite sides of line AC, so ∠CAE and ∠BCA are alternate angles formed by line AC.(1), (9), A9, D1
(15)∠CAE = ∠BCA(14), L1
(16)∠ABC + ∠BAC + ∠BCA = 180°(11), (13), (15), E
∎