I. ON ANGLES 17 Again turning the half-plane containing O around xy, the segment MO will fall on MO because the angles OMx and O Mx are right angles, and hence are equal. Since MO = MO, the point O falls on O , and therefore O and O coincide, as do the lines OO and OO . QED 19b. The reflection of a point O in a line xy is the endpoint of the perpendicular from O to the line, extended by the length of a segment equal to itself. It follows from the preceding considerations that this reflection is none other than the new position occupied by O after a rotation around xy. Given an arbitrary figure, we can take the reflection of each of its points. The set of these reflections constitutes a new figure, called the reflection of the first. We see that in order to obtain the reflection of a given figure in line xy, we can turn the plane of the figure around xy, so that each half-plane determined by the line falls on the other, then note the position taken by the original figure. It follows that: Theorem. A plane figure is congruent to its reflection. Corollary. The reflection of a line is a line. When a figure coincides with its reflection in line xy, we say that it is symmetric in this line, or that this line is an axis of symmetry of the figure. 20. To make a figure F coincide with its reflection F , we had to use a motion which took the figure out of its plane. We must note that this superposition is not possible without such a movement this holds because the sense of rotation is reversed in the two figures. We will now explain what this means. First, let us remark that the plane of the figure divides space into two regions. For brevity, let us call one of these the region situated above the plane the other, the region below the plane. Figure 17 Consider now an angle BAC in the figure F , which can be viewed as being described by a ray moving inside this angle from the position AB to the position AC (Fig. 17). Viewed from above this angle BAC will be said to have an inverse sense of rotation or a direct sense, according as the moving ray turns clockwise or counterclockwise.5 To be definite, let us consider the second situation. In this case, an observer lying along AB, with his feet at A, his head in the direction of B, and looking down, will see the side AC on his left and therefore, if still lying on AB, he faces AC, the region below the plane will be on his right. It is clear that to describe the sense of rotation of an angle viewed from below , this discussion can be repeated, with the word above replaced by the word below , and vice-versa. 5 We note that in order to define the sense of rotation, we must take into account the order of the sides. Thus angle BAC has the opposite sense from CAB.
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