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
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.
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
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 ,
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.