Angles about a point and the exterior angles of polygons
All angles around a point equal four right angles; a pentagon resolved into five triangles
Continuing his study of angle sums, Leonardo notes that all the angles made around a point are equal to four right angles. A pentagon is divided into five triangles meeting at its center, then redrawn with the central portions removed, since those central angles equal four right angles and can be discarded. Further figures show that the exterior angles of a polygon are equivalent to four right angles and that each exterior angle taken with its interior angle equals two right angles, proved by reference to the thirteenth proposition.
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Angles around a point equal four right angles
Three segments crossing at a single point demonstrate that all the angles made around a point are equal to four right angles. This is the lemma used to strip the central angles from the polygon decompositions that follow.
A pentagon resolved into five triangles
A pentagon is split into five triangles each with a corner at the center. Leonardo proves why, when the count is doubled and four are taken away, the number remains, then redraws the pentagon with the central pieces thrown away because they equal four right angles.
Exterior angles equal four right angles
With one vertex of a pentagon, square and triangle marked by dots, Leonardo states that the exterior angles are equivalent to four right angles, proved with the thirteenth, and that the exterior taken with the interior equals two right angles.