Exercise 35 Page 206

When the reflection of a figure in a line $m$ is followed by a second reflection in a line that is parallel to $m,$ line $k,$ this transformation is the same as a translation. According to the Reflections in Parallel Lines Theorem, if $A^{\prime \prime }$ is the image of $A$ we know the following. In this equation, $d$ is the distance between line $k$ and line $m.$ Therefore, to draw this, we need to know the distance between two corresponding vertices such as $A^{\prime \prime }$ and $A.$ For this purpose, we will draw a segment between $A^{\prime \prime }$ and $A$ using a straightedge.

To find the midpoint of $\overline{A{A}^{\prime \prime }},$ open a compass so that it's width is greater than half the length of the segment. Then, place the point of the compass at each endpoint and draw a pair of intersecting arcs.

The line that contains both intersections of the arcs, is the perpendicular bisector to $\overline{A{A}^{\prime \prime }}.$ We can draw it using a straightedge.

Next, adjust the compass so that it measures half of $A{A}^{\prime \prime }.$

Using this compass setting, we can mark any two points that are between $C^{\prime \prime }$ and $A$ on $\overline{A{A}^{\prime \prime }}.$ Through these points, we will draw our parallel lines.

In addition to being parallel, the lines also have to be perpendicular to $\overline{A{A}^{\prime \prime }}.$ To make this happen, draw two pairs of arcs around each point using an identical compass setting.

Open up the compass so that it's wider than the distance between any of the arcs and it's respective point. Then draw a pair of arcs above $X$ and $Y.$

By drawing a line through $X$ and $Z$ and another through $Y$ and $W,$ we create two parallel lines that are also perpendicular to $\overline{A{A}^{\prime \prime }}$ as well as $\overline{B{B}^{\prime \prime }}$ and $\overline{C{C}^{\prime \prime }}.$