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.
(1)A polygon satisfying these conditions is called a simple polygon.convention
(2)A simple polygon divides the plane into an inside and an outside. The interior angle at vertex A_i is the one of the two angles formed by the sides A_{i−1}A_i and A_iA_{i+1} that lies on the inside.(1), convention
(3)A simple polygon with n = 3 is a triangle.(1), convention
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Definition D2
A segment joining two non-adjacent vertices is called a diagonal. A polygon is called convex if the whole polygon lies on one side of the line containing any of its sides.
Proof.
(1)From one vertex of an n-gon, (n−3) diagonals can be drawn: subtract itself and its two neighbors.convention, A11
(2)In a convex polygon, every diagonal lies inside the polygon.convention
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Lemma L1
If k + m = n + 2, then 180(k−2) + 180(m−2) = 180(n−2)
Proof.
(1)180(k−2) + 180(m−2) = 180((k−2) + (m−2))A8
(2)(k−2) + (m−2) = (k + m) + ((−2) + (−2)) = (k + m) − 4A9, A10, A11
(3)Since k + m = n + 2, (k + m) − 4 = (n + 2) − 4 = n − 2(2), assumption, A9, A11, E
(4)180(k−2) + 180(m−2) = 180(n−2)(1), (2), (3), E
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Theorem T1
The interior angles of a convex n-gon sum to 180(n−2).
Proof.
(1)In the convex n-gon A₁A₂…A_n, draw the (n−3) diagonals joining A₁ to A₃, A₁ to A₄, …, A₁ to A_{n−1}.D2
(2)These diagonals all lie inside and do not meet one another, so they divide the polygon into (n−2) triangles A₁A₂A₃, A₁A₃A₄, …, A₁A_{n−1}A_n.(1), D2, A2
(3)The three interior angles of each triangle sum to 180°.(2), D1, A1
(4)Adding all interior angles of the (n−2) triangles gives 180(n−2)(2), (3), A7
(5)These angles are pieces that divide the polygon's interior angles without remainder. The angle at A₁ splits into (n−2) pieces, the angles at A₂ and A_n are undivided, and the angle at every other vertex splits into two pieces.(2), D1, D2
(6)At each vertex the sum of the pieces is the interior angle at that vertex.(5), A3
(7)Hence the interior angles sum to 180(n−2)(4), (6), E
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Theorem T2
Convexity is not required. The interior angles of every simple n-gon sum to 180(n−2).
Proof.
(1)In a non-convex polygon, a diagonal drawn from a vertex may go outside, so the decomposition of T1 cannot be applied as it stands.D2, T1
(2)For n = 3, the polygon is a triangle, so its interior angles sum to 180 = 180(3−2)D1, A1, A11
(3)Let n ≥ 4, and suppose the claim holds for every simple polygon with fewer than n sides.assumption
(4)This n-gon has a diagonal lying in its interior.(3), D1, D2, A4
(5)That diagonal divides the polygon into a simple k-gon and a simple m-gon with k + m = n + 2, 3 ≤ k < n, 3 ≤ m < n(4), A5
(6)By the induction hypothesis, the interior angle sums of the two pieces are 180(k−2) and 180(m−2).(3), (5)
(7)The interior angles of the two pieces divide the original polygon's interior angles without remainder. Only at the two endpoints of the diagonal does an interior angle split into two pieces.(5), D1
(8)At each vertex the sum of the pieces is the interior angle at that vertex, so adding the two pieces' angle sums gives the angle sum of the n-gon.(7), A3
(9)180(k−2) + 180(m−2) = 180(n−2)(5), L1
(10)Hence the interior angles of the n-gon sum to 180(n−2)(6), (8), (9), E
(11)The base case and the induction step are both verified, so the claim holds for every n ≥ 3.(2), (10), A6
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