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# hw5sol - Homework 5 1 If and m are lines which are not...

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Homework 5 1. If and m are lines which are not parallel, prove that there exists a point P and an angle θ so that ρ P,θ ( ) = m . Solution: Let P be the unique point of intersection of and m and let θ be the directed angle from to m . Then ρ P,θ maps to m . To see this, note that ρ P,θ sends P to P and for any other point Q on , it is mapped to a point Q 0 on m . Since isometries send lines to lines, it then follows that ρ P,θ ( ) = m as desired. 2. For any two lines ‘,m show that there is a line n so that σ n ( ) = m . Solution: If ‘,m are parallel, choose a line n parallel to ‘,m so that the directed distance from to n is exactly 1 2 the directed distance from to m . Now σ n ( ) = m (to check this, just note that any two points on map to m ). If ‘,m are not parallel and intersect ath the point P with directed angle from to m of θ , then let n be the line passing through P so that the directed angle from to n is

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hw5sol - Homework 5 1 If and m are lines which are not...

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