K-2. (a) Let l be a diameter of ? and let m be an open chord of ? that does not meet l and whose endpoints differ from the endpoints of l. Draw a diagram showing the common per- pendicular k to l and m in the Klein model. (Hint: Use the pole of m and the case 1 description of perpendicularity.) (b) Let l and m be intersecting open chords of ?. 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.
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### Part (a) - Drawing the common perpendicular k to / and m in the Klein model ** Show more…
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