Exercises and resolutions 292 Determine, preferably without using the formula, the sum of the internal angles of a: a) convex pentagon; b) convex hexagon. Ver respostaa) $540^{\circ}$ · b) $720^{\circ}$ — decomposing into 3 and 4 triangles, respectively. 293 Determine the value of $x$ in cases (in and), $\overline{AB} \parallel \overline{ED}$): Ver respostaa) $70^{\circ}$ · b) $110^{\circ}$ · c) $90^{\circ}$ · d) $120^{\circ}$ · e) $120^{\circ}$
How many diagonals does a polygon have? How much do your angles add up? The answers come from a simple idea: dividing the polygon into triangles. Who discovered The sum of the internal angles of a polygon is an extension of the proposition I.32 of The Elements of Euclid, which deals with the triangle. Proclus (5th century AD) recorded the two consequences for convex polygons: the sum of the interior angles of a polygon with $n$ sides is equal to $2n - 4$ right angles, and the sum of the exterior angles is always equal to four right angles. In 1619, Johannes Kepler used these sums to figure out which regular polygons can tile the plane.
Triangles, squares, pentagons, hexagons: polygons are in honeycombs, road signs and screw nuts. Who discovered The simplest regular polygons were constructed with a ruler and compass by Euclid (c. 300 BC), in book IV of The Elements: triangle, square, pentagon, hexagon and the polygon of 15 sides. For more than two thousand years, no one has managed to construct another regular polygon with a prime number of sides. In 1796, at the 18 age, Carl Friedrich Gauss constructed the heptadecagon, the regular polygon with 17 sides.