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.
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.
The bisectors of B and C form an angle equal to
A + one right angle.
The bisectors of the exterior angles of B and C form an angle equal to one
right angle −
25. In a convex quadrilateral:
The bisectors of two consecutive angles form an angle equal to one half the
sum of the other two angles.
The bisectors of two opposite angles form a supplementary angle to the
half the diﬀerence of the other two angles.