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.
(1)Adjacent sides do not lie on one line. If they did, that point would not be a vertex.convention
(2)Hence A_{i+1} is not on line A_{i−1}A_i.(1)
(3)By convexity, the interior angle at A_i is the very angle ∠A_{i−1}A_iA_{i+1} formed by the sides A_{i−1}A_i and A_iA_{i+1}, and it is less than 180°.convention, A5
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Definition D2
Let E_i be a point on the ray extending side A_{i−1}A_i beyond A_i. The angle ∠A_{i+1}A_iE_i is called the exterior angle at vertex A_i.
Proof.
(1)Such a point E_i exists.A4
(2)The angle obtained by extending the other side A_iA_{i+1} is the vertical angle of this one, hence has the same size. Whichever side is extended, the size of the exterior angle is uniquely determined.(1), A7
(3)Take one exterior angle at each vertex, obtaining n of them.(2), D1
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Lemma L1
At each vertex the interior and exterior angles sum to 180°.
Proof.
(1)A_{i−1}, A_i, E_i are on one line with A_i between, so ∠A_{i−1}A_iE_i = 180°D2, A1
(2)A_{i+1} is not on line A_{i−1}A_i, so ray A_iA_{i+1} lies in the interior of that straight angle.D1, A3
(3)∠A_{i−1}A_iA_{i+1} + ∠A_{i+1}A_iE_i = ∠A_{i−1}A_iE_i = 180°(1), (2), A2
(4)The former angle is the interior angle and the latter the exterior angle, so interior + exterior = 180°(3), D1, D2
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Theorem T
The exterior angles of a convex n-gon sum to 360°.
Proof.
(1)At each of the vertices A₁ through A_n, interior + exterior = 180°L1, D1
(2)Adding all n equations gives (sum of interior angles) + (sum of exterior angles) = 180n(1), A8, A10
(3)sum of interior angles = 180(n−2)D1, A6
(4)180(n−2) + (sum of exterior angles) = 180n(2), (3), E
(5)180(n−2) = 180n + 180(−2) = 180n − 360A9, A12, A14
(6)180(n−2) + 360 = (180n − 360) + 360 = 180n(5), A10, A12, A13, E
(7)180(n−2) + (sum of exterior angles) = 180(n−2) + 360(4), (6), E
(8)sum of exterior angles = 360°(7), A10, A11
(9)180n came from the number of vertices, and 180(n−2) came from the sum of interior angles. Comparing the two, the terms containing n cancel each other, and only the constant 360 remains. The reason the sum of exterior angles is independent of the number of sides is that cancellation.(2), (3), (8)
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