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.

5We

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.