Example 8.  Constructing a rectangle (with an introduction to constructing parallel lines).

In this example we construct a rectangle.  Remember that a rectangle is a quadrilateral with two sets of parallel congruent sides and four congruent angles.  As with a square, each angle measurement is .  Let's begin by constructing the base of the rectangle.  Construct segment AB.  Next, construct a perpendicular line to AB through A and another through B.  Remember that two perpendicular lines to the same line (or segment) are parallel.  So these two lines are parallel.  Next construct a point on the perpendicular through B. (It really does not matter which line you use; this is just for uniformity of directions.)  Label the point C.  Drag C anywhere you want to.  Although we could use perpendicular lines to complete the rectangle, we will use parallel lines instead.  Select segment AB and point C.  Select the Parallel Line command in the Construct menu.  See Figure 33.

Figure 33

Find the intersection of this line with the perpendicular line to AB through A.  This intersection is the fourth vertex of the rectangle.  Label it D.  Now, hide the lines so that only segment AB and vertices C and D are left.  Construct segments BC, CD and DA.  See Figure 34.

Figure 34

An alternative way to construct the segments without using the Toolbox follows.  Select two vertices.  Open the Construct menu and select Segment.  Actually this method works nicely when constructing polygons.  Select all the vertices, B, C, D and A in that order (or clockwise) and select the Segment command in the Construct menu.  This should complete the rectangle.  Now perform the drag test.  Only vertices A, B and C can be dragged since D depends on C and segment AB.  A and B determine the length of the one side and C determines BC or CD, the length of the other.  Measure opposite sides, say AB and CD.  Are they congruent?  Figure 35 shows a possible rectangle and the measurement of two opposite sides and three angles.