Review: Angles on a Line
      Use the diagram at 
      the right to find the                     (x+8)°
      values of x, y and 
      m∠4                                  1    2 3(2x-8)°
                                        y° 6    5      4
                                              (4x-24)°
     Review: Angles on a Line 
     (Solution)
     m∠1 + m∠2 + m∠3 = 180°
     90 + (x + 8) + (2x - 8) = 180
     3x + 90 = 180                                (x+8)°
     3x = 90
     x = 30                                  1    2    (2x-8)°
                                                      3
     m∠6 = m∠3 (By vertical angles)          6            4
     y = 2x – 8                           y°      5
     y = 2(30) – 8                              (4x-24)°
     y = 52
     m∠4 + m∠5 + m∠6 = 180°
     m∠4 + (4x - 24) + y = 180
     m∠4 + 4(30) – 24 + 52 = 180
     m∠4 + 148 = 180
     m∠4 = 32°
    The Auxiliary Line
     Auxiliary Line (Helping Line): an extra line 
      needed to complete a proof/problem in 
      plane geometry.
                          1
                            1
                            2
                           2
                      Recall: Through any point there 
                      is exactly one line that will be 
                      parallel to an already existing 
                      line!
     Example #1: Auxiliary Line
                             These two angles are    144°
                             consecutive interior 
     m∠1 = 36°+56°           angles so they sum to            36°
     m∠1 = 92°               180°                                 1
                                                              56°
                              These two angles are 
                              alternate interior angles 
                              so they are the same
                                                         56°
   Example #2: Auxiliary Line
   Both of the angle 
   measures are found using 
   consecutive interior 
   angles theorem           1
                         18° 82°
   m∠1 = 18°+82°   162°
   m∠1 = 100°                  98°