44 V. PARALLEL LINES number of sides minus two because if A is taken as the common vertex of these triangles, all the other sides of the polygon are the sides opposite A, except the two sides which end at A. The sum of the angles of these triangles gives us the sum of the angles of the polygon. The theorem is proved. If n is the number of sides of the polygon, the sum of the angles is 2(n 2) or 2n 4 right angles. Corollary. The sum of the exterior angles of a convex polygon, formed by extending the sides in the same sense, is equal to four right angles. Indeed, an interior angle plus the adjacent exterior angle gives us two right angles. Adding the results for all the n vertices we obtain 2n right angles, of which 2n 4 are given by the sum of the interior angle. The sum of the exterior angles is equal to the four missing right angles. Exercises Exercise 21. In a triangle ABC, we draw a parallel to BC through the in- tersection point of the bisectors of B and C. This parallel intersects AB in M and BC in N. Show that MN = BM + CN. What happens to this statement if the parallel is drawn through the intersection point of the bisectors of the exterior angles at B and C? Or through the intersection of the bisector of B with the bisector of the exterior angle at C? Sum of the angles of a polygon. Exercise 22. Prove the theorem in 44b by decomposing the polygon into triangles using segments starting from an interior point of the polygon. Exercise 23. In triangle ABC we draw lines AD, AE from point A to side BC, such that the first makes an angle equal to C with AB, while the second makes an angle equal to B with AC. Show that triangle ADE is isosceles. Exercise 24. In any triangle ABC: 1◦. The bisector of A and the altitude from A make an angle equal to half the difference between B and C. 2◦. The bisectors of B and C form an angle equal to 1 2 A + one right angle. 3◦. The bisectors of the exterior angles of B and C form an angle equal to one right angle 1 2 A. 25. In a convex quadrilateral: 1◦. The bisectors of two consecutive angles form an angle equal to one half the sum of the other two angles. 2◦. The bisectors of two opposite angles form a supplementary angle to the half the difference of the other two angles.
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