PROOF 147

Prove.

The exterior angles always sum to 360°

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
A3interior of a straight angle = one half-plane
the interior of the straight angle ∠A_{i−1}A_iE_i is one side of line A_{i−1}A_i
A4points on both sides of a point on a line
Euclid's second postulate — a segment can be extended indefinitely
A5interior angle of a convex polygon < 180°
convexity — the interior angle is the very angle formed by two adjacent sides
A6sum of interior angles = 180(n−2)
Proof 146 — the triangle angle sum, hence stands on the parallel postulate
A7vertical angles are equal
Proof 48 — whichever side is extended, the exterior angle has the same size
A8adding a n times gives na
natural multiples
A9a(b+c) = ab + ac
distributive law
A10(a+b)+c=a+(b+c), a+b=b+a
associativity and commutativity of addition
A11a + c = b + c ⟹ a = b
cancellation law of addition
A12a−b=a+(−b), a(−b)=−(ab)
convention of subtraction and sign rules
A13a + (−a) = 0, a + 0 = a
additive inverse and identity
A14180 × 2 = 360
numerical computation
Ea = b ⟹ same operation on both sides
properties of equality

Nothing outside this list may be cited.

Definition D1

The convex n-gon A₁A₂…A_n is a simple polygon of n points and n sides, and the whole polygon lies on one side of the line containing any of its sides. Indices are read with A_{n+1} = A₁, A₀ = A_n.

Proof.

Related labs

Polygons lab →