Consider a and one exterior angle at each vertex. Let S be the sum of the measures of the exterior angles.
Notice that
∠3 lies inside the polygon. In this case, its measure is considered to be . Because of this,
∠3 and
∠C are supplementary.
m∠C+m∠3=180∘(I)
The rest of exterior angles form a linear pair with their corresponding interior angles. This means all those pair of angles are supplementary.
⎩⎪⎪⎪⎪⎨⎪⎪⎪⎪⎧m∠1+m∠A=180∘m∠2+m∠B=180∘m∠4+m∠D=180∘m∠5+m∠E=180∘
Next, add these four equations to Equation (I). Remember that the Polygon Interior Angles Theorem states that the sum of the measures of the interior angles of a polygon with
n sides is
(n−2)⋅180∘.
m∠1+m∠Am∠2+m∠Bm∠3+m∠Cm∠4+m∠D+m∠5+m∠ES+(5−2)⋅180∘=180∘=180∘=180∘=180∘=180∘=5⋅180∘
Finally, solve the resulting equation for
S.
S+(5−2)⋅180∘=5⋅180∘
S+3⋅180∘=5⋅180∘
S+540∘=900∘
S=360∘
As can be seen, the sum of the measures of the exterior angles of the concave polygon is
360∘, which completes the proof. Keep in mind that the same reasoning can be applied to any other concave polygon.