The symmetries preserve the local environment of the line. Notice how in reflection and half-turn symmetry the line and its local environment are both part of the symmetry action and how the relationship between them is integral to the action. In fact, reflection in the line does not move the line at all but exhibits a way in which the space on the two sides of the line are the same. An isometry is a transformation of a region of space that preserves distances and angle measures. You will show in Problem 18.3 that every isometry is either a translation, a rotation, a reflection, or a composition of them.
Some symmetries hold only for straight lines, while some work with other curves too.
Try to determine which ones are specific to straight lines and why. Also think of practical applications of these symmetries for constructing a straight line or for determining if a line is straight. See Figure 1.3. They then hold the boards up to the light. If the edges are not straight, there will be gaps between the boards which light will shine through.