Perpendicular Lines

1 Key excerpts on "Perpendicular Lines"

• 

  eBook - ePub

  Introduction to Non-Euclidean Geometry

  • Harold E. Wolfe(Author)
  • 2013(Publication Date)
  • Dover Publications

    (Publisher)

  But two lines do not need to be intersecting in order to have a common parallel. A common parallel can be constructed for any two lines. If they do not intersect, one has only to draw through any point of the first a parallel to the second and then construct a common parallel to the pair of intersecting lines. Obviously two parallel lines have only one common parallel which is parallel to them in opposite directions.

  Finally, interpreted otherwise, this construction of a line parallel to two lines in given senses enables us to effect the construction of the line joining any two given ideal points. 51. The Construction of a Line Perpendicular to One of Two Intersecting Lines and Parallel to the Other.

  We return now to the problem of constructing a line perpendicular to one of two intersecting lines and parallel to the other, or, otherwise, of constructing the distance corresponding to any acute angle regarded as an angle of parallelism. This construction is readily accomplished by the use of the results of the last section.

    Figure 49

  Given the acute angle ABC (Fig. 49 ), we wish to construct a line perpendicular to BA and parallel to BC. All that is needed is to construct the angle ABD equal to angle ABC. Then the common parallel to BC and BD will be perpendicular to BA and parallel to BC. This construction can always be made, whatever the size of the given acute angle, no matter how small or how near a right angle.

  Here again attention is called to the generality of the construction. A line can be constructed perpendicular to one of two lines and parallel to the other even when they do not intersect, whether they be parallel or non-intersecting. The modification of the construction for these cases has already been suggested.

  EXERCISE

  How many lines can be constructed which are perpendicular to one of two given lines and parallel to the other, if the given lines (a) intersect at an acute angle? (b) are perpendicular? (c) are parallel? (d) are non-intersecting?

  52. Units of Length and Angle.

  We speak of the units which we use for measuring angles as being absolute

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