PROOF 145

Prove.

Alternate and corresponding angles at parallel lines are equal

On the Euclidean plane.

What you may use

A1O between A,B ⟹ ∠AOB=180°
straight angle
A2∠XOP + ∠POY = ∠XOY
additivity of angles — when ray OP lies in the interior of ∠XOY
A3a point off a line lies on one side
plane separation — a line divides the plane into two half-planes
A4interior of a straight angle = one half-plane
the interior of the straight angle ∠APA′ is one side of line AA′
A5one line through two points
Euclid's first postulate
A6points on both sides of a point on a line
Euclid's second postulate — a segment can be extended indefinitely
A7one angle of size θ per half-plane
construction of an angle (Euclid I.23) — unique on a given side of a given ray
A8vertical angles are equal
Proof 48 — does not use the fifth postulate
A9exterior angle > non-adjacent interior angle
Euclid I.16 — does not use the fifth postulate
A10P ∉ ℓ ⟹ exactly one parallel
parallel postulate — Playfair's form (equivalent to the fifth postulate)
A11a + c = b + c ⟹ a = b
cancellation law of addition
A12a + b = b + a
commutativity of addition
A13a > b ⟹ a ≠ b
trichotomy of order
Ea = b ⟹ same operation on both sides
properties of equality

Nothing outside this list may be cited.

Definition D1

A line t cuts two distinct lines ℓ, m at P and Q respectively. On ℓ take two points A, A′ with P between them; on m take two points B, B′ with Q between them; on t take a point S so that P lies between S and Q; and assign the names so that A and B lie on opposite sides of line t.

Proof.

Related labs

Angles lab →