PROOF 146

Prove.

The interior angles of an n-gon sum to 180(n−2)

On the Euclidean plane.

What you may use

A1the interior angles of a triangle sum to 180°
Proof 49 — equivalent to the parallel postulate
A2convex n-gon → n−2 triangles
decomposition of a convex polygon — the (n−3) diagonals from one vertex all lie in the interior
A3∠XOP + ∠POY = ∠XOY
additivity of angles — when ray OP lies in the interior of ∠XOY
A4n ≥ 4 ⟹ an interior diagonal exists
a simple polygon has at least one diagonal lying in its interior
A5n-gon = k-gon + (n+2−k)-gon
splitting by an interior diagonal — both pieces are simple polygons with fewer than n sides
A6P(3), all earlier ⟹ P(n)
strong mathematical induction
A7adding a k times gives ka
natural multiples
A8a(b+c) = ab + ac
distributive law
A9(a+b)+c = a+(b+c)
associativity of addition
A10a + b = b + a
commutativity of addition
A11a−b = a+(−b), a+(−a) = 0
convention of subtraction and additive inverse
Ea = b ⟹ same operation on both sides
properties of equality

Nothing outside this list may be cited.

Definition D1

For n ≥ 3, the n-gon A₁A₂…A_n consists of n points and n segments A₁A₂, A₂A₃, …, A_nA₁. Non-adjacent sides do not meet, and adjacent sides do not lie on one line.

Proof.

Related labs

Polygons lab →