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					    Geometry

Lines and Angles


      CONFIDENTIAL   1
                  Warm Up
     Find the circumference and area of each circle.



1)                             2)
             80 cm                       3.8 m




                      CONFIDENTIAL                     2
 Parallel, perpendicular and skew lines




Pairs of lines can relate to each other in four different
ways: intersecting lines, parallel lines, perpendicular
 lines and skew lines. These concepts are useful for
     understanding and solving various geometry
                        problems.




                      CONFIDENTIAL                          3
  Parallel, perpendicular and skew lines


Parallel lines (||) are lines that never intersect i.e. they
  are coplanar. The distance between the two lines is
   fixed and the two lines go in the same direction.
           In the figure, AB || EF and EG || FH.


                     A                   B

                 E                F

                      C                  D

                 G                H


                          CONFIDENTIAL                         4
 Parallel, perpendicular and skew lines


Perpendicular lines (|)are lines that intersect at one point
 and form a 90° angle i.e. two different straight lines on
the same plane in two different directions who meet each
 other at only right angles are called Perpendicular Lines.
            In the figure, AB | AE and EG | GH.


                    A                  B

                E               F

                    C                  D

                G               H

                        CONFIDENTIAL                           5
  Parallel, perpendicular and skew lines


  Skew lines are not coplanar. Skew lines only happen in
space. Skew lines never intersect because they are not on
 the same plane. Skew lines are difficult to draw because
        they exist in the three dimensional space.
           In the figure, AB and EG are skew.


                    A                  B

                E               F

                    C                  D

                G               H

                        CONFIDENTIAL                        6
Parallel, perpendicular and skew lines


 Parallel planes are planes that do not intersect.
     In the figure, plane ABE || plane CDG.



                  A                B

              E              F

                  C                D

              G              H



                   CONFIDENTIAL                      7
       Identifying types of lines and planes
                 Identify each of the following:


A) A pair of parallel segments.
  KN || PS
                                              K        L
B) A pair of skew segments.
  LM || RS                                N        M


C) A pair of perpendicular segments.          P        Q
  MR | RS
                                          S        R

D) A pair of parallel planes.
  plane KPS || plane LQR.

                           CONFIDENTIAL                    8
 Referring to the figure, we can conclude:




AB is perpendicular to CL and CIB is 90°
•KE intersects IB at point J
•GH is parallel to AB
•KD is perpendicular to MH and KLH is 90°
•IL intersects JM at point K
•EF intersects GL at point M, intersects IL at point K and IB at point J




                              CONFIDENTIAL                         9
                       Now you try!

               Identify each of the following:


                                             C       D
1a) A pair of parallel segments
                                         B       E
1b) A pair of skew segments

1c) A pair of perpendicular segments         G       H

1d) A pair of parallel planes            F       J




                          CONFIDENTIAL               10
     Angle pairs formed by a transversal


A transversal is a line that intersects two coplanar lines at
  two different points. The traversal t and the other two
              lines r and s form eight angles.



                             exterior angles


                             interior angles


                                exterior angles



                       CONFIDENTIAL                         11
 Corresponding angles are created where a transversal
crosses other (usually parallel) lines. The corresponding
   angles are the ones at the same location at each
         intersection i.e. angle 1 and angle 5.




Corresponding angles




                      CONFIDENTIAL                          12
Alternate interior angles are created where a transversal
 crosses two (usually parallel) lines. Each pair of these
angles are inside the parallel lines, and on opposite sides
       of the transversal i.e. angle 3 and angle 5.




                           Alternate interior angles




                      CONFIDENTIAL                            13
 Alternate exterior angles are created where a transversal
   crosses two (usually parallel) lines. Each pair of these
angles are outside the parallel lines, and on opposite sides
        of the transversal i.e. angle 1 and angle 7.




                             Alternate exterior angles




                       CONFIDENTIAL                        14
   Same side interior angles crosses two (usually
parallel) lines. Each pair of interior angles are inside
   the parallel lines, and on the same side of the
        transversal. i.e. angle 3 and angle 6.




                                Same side interior angles




                     CONFIDENTIAL                          15
        Classifying pairs of angles
        Give an example of each angle pair.


A) Corresponding angles
   Angle 4 and angle 8

B) Alternate interior angles
    Angle 4 and angle 6

C) Alternate exterior angles
    Angle 2 and angle 8

D) Same side interior angles
    Angle 4 and angle 5



                    CONFIDENTIAL              16
         Now you try!

Give an example of each angle pair.

   2a) Corresponding angles

   2b) Alternate interior angles

   2c) Alternate exterior angles

   2d) Same side interior angles




           CONFIDENTIAL               17
      Identifying angle pairs and transversals
      Identify the transversal and classify each angle pair.


A) Angle 1 and angle 5
transversal: n; Alternate interior angles
                                                            4
                                                                 l

B) Angle 3 and angle 6                                  5
transversal: m; Corresponding angles
                                                    6


C) Angle 1 and angle 4                          m           n
transversal: l; Alternate exterior angles




                           CONFIDENTIAL                         18
                     Now you try!

3) Identify the transversal and classify the angle pair   2
                  and 5 in the diagram.




                                      4
                                          l


                                 5
                             6


                         m            n



                       CONFIDENTIAL                           19
                       Assessment

               Identify each of the following:


                                             B       C
1) A pair of perpendicular segments
                                         A       D
2) A pair of skew segments

3) A pair of parallel segments               F       G

4) A pair of parallel planes             E       H




                          CONFIDENTIAL               20
Give an example of each angle pair.

   5) Alternate interior angles

   6) Alternate exterior angles

   7) Corresponding angles

   8) Same side interior angles




           CONFIDENTIAL               21
Identify the transversal and classify each angle pair.



                                                     m
9) Angle 1 and angle 2
                                                         n
10) Angle 2 and angle 3
                                        4
11) Angle 2 and angle 4                     5

12) Angle 4 and angle 5                     3
                                                 p




                     CONFIDENTIAL                            22
                      Let’s review

  Parallel, perpendicular and skew lines


Parallel lines (||) are lines that never intersect i.e. they
  are coplanar. The distance between the two lines is
   fixed and the two lines go in the same direction.
           In the figure, AB || EF and EG || FH.


                     A                   B

                 E                F

                      C                  D

                 G                H

                          CONFIDENTIAL                         23
 Parallel, perpendicular and skew lines


Perpendicular lines (|)are lines that intersect at one point
 and form a 90° angle i.e. two different straight lines on
the same plane in two different directions who meet each
 other at only right angles are called Perpendicular Lines.
            In the figure, AB | AE and EG | GH.


                    A                  B

                E               F

                    C                  D

                G               H

                        CONFIDENTIAL                       24
  Parallel, perpendicular and skew lines


  Skew lines are not coplanar. Skew lines only happen in
space. Skew lines never intersect because they are not on
 the same plane. Skew lines are difficult to draw because
        they exist in the three dimensional space.
           In the figure, AB and EG are skew.


                    A                  B

                E               F

                    C                  D

                G               H

                        CONFIDENTIAL                        25
Parallel, perpendicular and skew lines


 Parallel planes are planes that do not intersect.
     In the figure, plane ABE || plane CDG.



                  A                B

              E              F

                  C                D

              G              H



                   CONFIDENTIAL                      26
     Angle pairs formed by a transversal


A transversal is a line that intersects two coplanar lines at
  two different points. The traversal t and the other two
              lines r and s form eight angles.



                             exterior angles


                             interior angles


                                exterior angles



                       CONFIDENTIAL                         27
 Corresponding angles are created where a transversal
crosses other (usually parallel) lines. The corresponding
   angles are the ones at the same location at each
         intersection i.e. angle 1 and angle 5.




Corresponding angles




                      CONFIDENTIAL                          28
Alternate interior angles are created where a transversal
 crosses two (usually parallel) lines. Each pair of these
angles are inside the parallel lines, and on opposite sides
       of the transversal i.e. angle 3 and angle 5.




                           Alternate interior angles




                      CONFIDENTIAL                            29
 Alternate exterior angles are created where a transversal
   crosses two (usually parallel) lines. Each pair of these
angles are outside the parallel lines, and on opposite sides
        of the transversal i.e. angle 1 and angle 7.




                             Alternate exterior angles




                       CONFIDENTIAL                        30
   Same side interior angles crosses two (usually
parallel) lines. Each pair of interior angles are inside
   the parallel lines, and on the same side of the
        transversal. i.e. angle 3 and angle 6.




                                Same side interior angles




                     CONFIDENTIAL                          31
You did a great job
      today!




      CONFIDENTIAL    32

				
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