PROOF 49

Prove.

The angles of a triangle sum to 180°

On the Euclidean plane.

What you may use

A1O between A,B ⟹ ∠AOB=180°
straight angle
A2P ∉ ℓ ⟹ exactly one parallel
parallel postulate — Playfair's axiom (equivalent to the fifth postulate)
A3equal alternate angles ⟹ ℓ ∥ m
Euclid's Proposition 27 — does not use the parallel postulate
A4one angle of size θ per half-plane
angle construction — the angle of a given size on a given side of a given ray is unique
A5∠XOP + ∠POY = ∠XOY
additivity of angles — when ray OP is interior to ∠XOY
A6PQ ∩ ℓ = ∅ ⟺ same side
plane separation — a line divides the plane into two sides
A7interior = intersection of two half-planes
interior of an angle — the interior of ∠XOY is the points on the Y side of line OX and the X side of line OY
A8interior of a straight angle = one half-plane
the interior of the straight angle ∠DAE is one side of line DE
A9P interior ⟹ X, Y are separated
if ray OP is interior to ∠XOY, then X and Y lie on opposite sides of line OP
A10points on both sides of a point on a line
Euclid's second postulate — a segment can be extended indefinitely
Ea = b ⟹ same operation on both sides
properties of equality

Nothing outside this list may be cited.

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.

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