## Chapter 6, Major Exercise 6

Above is an interactive version of figure 6.32. You can drag any of the red
points around using the mouse.

Recall, you are to show that if lines *l* and *m* (green)
are parallel but not limiting parallel,
we construct *AA'* and *BB'* (purple) to be perpendicular
to *m*. Then, assuming that *AA'* is longer than *BB'*, we
can create ray *EF* (dark green)
so that *EA'* is congruent to *BB'*, and so that
angles *A'EF* and *B'BG* are congruent.

You are to show that *EF* intersects line *l* at a point *H*.

In order to do this exercise, you may use Major exercises 2, 3, 4, and 5.
Indeed, you almost certainly *must* use them.

This exercise is needed to show that Hilbert's construction of the mutual perpendicular to lines

*l*and

*m*exists. Below is that construction, which works as follows. (see page 263 of the text).

First, we create ray *EF* as above, and from the exercise, we
know it intersects *AB* at a point *H*. Now we find the
unique point *K* on ray *AB* so that *EH* is congruent
to *BK*. Drop perpendiculars *HH'* and *KK'*,
and observe that they are congruent (since Lambert
quadrilaterals *A'H'HE* and *B'K'KB* are congruent).

Since we have two congruent sides perpendicular to the base,
quadrilateral *H'K'KH* is a Saccheri quadrilateral.
We showed earlier that the line joining the midpoint *M*
of the summit *HK* to the midpoint *M'* of the base *H'K'*
will be perpendicular to both lines.

Created with Cinderella

Scott Sutherland, Apr 28 2009.