18 I. ON ANGLES Since an observer lying on AB and facing AC will necessarily have the region above the plane on his left, if the region below is on his right, and vice-versa, we see that the sense of rotation changes depending on whether we view the angle from one side of the plane or from the other.6 Let us now suppose that an angle is moved in any way at all, without ever leaving the plane. An observer moving along with the angle will not change his position relative to the regions of space above and below the plane, and therefore the sense of rotation is not altered by a motion which does not leave the plane. To prove that such a motion cannot make a figure F coincide with its reflection F , it suffices therefore to see that the senses of rotation of the two figures are opposite. But we have seen that we can make F coincide with F by turning the plane around xy (19). As a result of this rotation, the points above the plane are moved below it, and vice-versa. The sense of rotation of an angle in F , viewed from above, is therefore the same as the sense of F viewed from below, so that the two angles, viewed from the same side, have opposite senses of rotation. QED 20b. Remarks. I. We say that a plane is oriented when a sense of rotation for angles has been chosen as the direct sense. According to the preceding paragraph, orienting a plane amounts to deciding which region of space will be said to be above the plane. II. It is convenient to regard an angle with vertex O as being described by a ray starting from O which at first coincides with the first side, and then turns about O in the plane (in the direct or inverse sense) until it coincides with the second side. If the ray has made a quarter of a complete turn, the angle is a right angle. If it has made one half of a complete turn, the angle is a straight angle, such as AOA (Figures 11 or 13). Nothing prevents us, by the way, from considering angles greater than two right angles, since our ray can make more than half a complete turn. III. Clearly, an arc of a circle, like an angle, can have a direct or an inverse sense, which depends, of course, on the order in which one gives its endpoints. Two points A, B on a circle divide it into two different arcs of which (unless the two points are diametrically opposite) one is a minor arc (less than a semicircle), and the other a major arc. It must be noted that, since the endpoints A, B are given in a certain order (for example if A is first and B is second), these two arcs have opposite senses. Exercises Exercise 1. Given a segment AB and its midpoint M , show that the distance CM is one half the difference between CA and CB if C is a point on the segment. If C is on line AB, but not between A and B, then CM is one half the sum of CA and CB. Exercise 2. Given an angle AOB, and its bisector OM , show that angle COM is one half the difference of COA and COB if ray OC is inside angle AOB it is the supplement of half the difference if ray OC is inside angle A OB which is vertical to AOB it is one half the sum of COA and COB if OM is inside one of the other angles AOA and BOB formed by these lines. 6 For this reason, writing viewed through a transparent sheet of paper appears reversed.
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