exercises, on the Klein model; Poincaré models and on circles; (3) H-exercises, on harmonic tetrads and theorems of Menelaus, Ceva, Gergonne, and Desargues; (4) projects. The K-exercises and P-exercises are extremely important for a visual understanding of plane hyperbolic geometry. K-Exercises K-1. Verify the interpretations of the incidence axioms, the between- ness axioms, and Dedekind's axiom (if the Euclidean plane is real) for the Klein model. K-2. (a) Let I be a diameter of y and let m be an open chord of y that does not meet 1 and whose endpoints differ from the endpoints of 1. Draw a diagram showing the common per- pendicular k to land m in the Klein model. (Hint: Use the pole of m and the case 1 description of perpendicularity.) (b) Let I and m be intersecting open chords of y. It is a valid theorem in hyperbolic geometry that for any two intersect- ing nonperpendicular lines there exists a third line perpen- dicular to one of them and asymptotically parallel to the other (see Major Exercise 9, Chapter 6). Draw the two lines in the Klein model that are perpendicular to l and asymp- totically parallel to m (on the left and right, respectively). This shows that the angle of parallelism can be any acute angle whatever. Explain. (c) In the Euclidean plane, any three parallel lines have a com- mon transversal. Draw three parallel lines in the Klein model that do not have a common transversal.to al (21) K-3. (a) In the Klein model, an ideal point and an ordinary point al- ways determine a unique Klein line. Translate this back into a theorem in hyperbolic geometry about limiting parallel rays. (b) Suppose the ultra-ideal points P(I) and P(m) are poles of Klein lines I and m, respectively. You saw in Figure 7.18 that the Euclidean line joining P(O