Exercise 29 Page 526

A square is a rectangle with four congruent sides.

To construct a square with a side length of $AB,$ we will begin by drawing a horizontal segment that's longer than $\overline{AB}.$

Let's use a compass to make these segments have the same length. We first need to adjust the compass so that its ends are aligned with the endpoints of $\overline{AB}.$ Then, we will move the compass to the new segment and mark where the end should be with the pencil.

Next, we will draw our first vertical side. To do so, we will place the sharp point of the compass at the point we marked in the previous step and, using the same compass setting, we will draw two arcs around it.

Let's now open the compass so that it's greater than the distance between the first mark and either arc. Then, we will place the point at each point of intersection between the two arcs and the line, and draw two more arcs that intersect above (or below).

Next, we will use a straightedge to draw the first vertical side of our square.

To make the vertical side have the same length as $\overline{AB},$ we need to repeat the procedure from when we measured our horizontal side.

Next, we will draw our second horizontal side. Let's repeat the procedure we used to draw the vertical side.

Again, we have to measure the second horizontal side so that it has the same length as $\overline{AB}.$

Finally, we will draw the last vertical side using a straightedge.