I. ON ANGLES 11 Indeed, let us turn (10) the angle BOA onto itself (Fig. 10). The side OB will take the place of OA , and, since 0A will take the former position of OB, this means that ray OA, which is the extension of OA , will coincide with OB , the extension of OB. Thus angle AOB is superimposed on angle A OB , and therefore these two angles are congruent. 13. Arcs and angles. Every ray issuing from the center of a circle meets this circle in one and only one point. Every angle AOB (Fig. 4) with its vertex at the center O of a circle determines an arc AB having as endpoints the intersections of the sides of the angle with the circle. In general, (see however 20b), we consider the arc which is less than a semicircle. Conversely, every arc less than a semicircle can be thought of as determined by an angle with its vertex at the center of the circle, formed by the rays passing through the endpoints of the arc. Theorem. On the same circle, or on equal circles: 1◦. To equal arcs (smaller than a semicircle) there correspond equal central angles 2 2◦. To unequal arcs (smaller than a semicircle) there correspond unequal central angles, and the greater angle corresponds to the greater arc 3◦. If an arc (smaller than a semicircle) is the sum of two others, the corre- sponding central angle is the sum of the angles associated with the smaller arcs. 1◦, 2◦. Let AB, AC (Fig. 4) be the two arcs, drawn on the same circle, starting from the same point A, in the same direction (8). The two angles AOB, AOC are therefore placed as in 11. But then rays OA, OB, OC are placed in the same order as points A, B, C on the circle. Moreover, if rays OB, OC coincide, then so do the points B, C, and conversely. 3◦. Let us recall that the sum of two arcs (8b) is formed by placing them as arcs AB, BC are placed (Fig. 4). Then the angle AOC corresponding to the sum of these arcs will be the sum of AOB and BOC, because these angles are adjacent. According to this result, in order to compare angles, one can draw circles with the same radius, centered at the vertices of the angles, and compare the arcs inter- cepted on these circles. The division of angles into two or more equal parts corresponds to the division into equal parts of the corresponding arc of a circle centered at the vertex of the angle. 14. Perpendiculars. Right angles. We say that two lines are perpendicular to each other if, among the four angles they form, two adjacent angles are equal to each other. For example, line AOA (Fig. 11) is perpendicular to BOB if the angles numbered 1 and 2 in the figure are equal. In such a case, the four angles at O are equal to each other, since angles 3 and 4 (Fig. 11) are equal respectively to 1 and 2, because they are vertical angles. An angle whose sides are perpendicular is called a right angle. 2 This is the standard term in American texts. Hadamard uses the locution ‘angles with their vertex at the center’ or ‘angles at the center’. –transl.
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