1.2 Points_ Lines and Planes

					1.1 Points, Lines and
       Planes
  Honors Modern Geometry
         Mr. Sgroi
Using Undefined terms and
definition
A definition uses known words to describe a
  new word. In geometry, some words
  such as point, line and plane are
  undefined terms or not formally defined.
Using Undefined terms and
definition

• A point has no
  dimension. It is        A

  usually represented
                        Point A
  by a small dot.
Using Undefined terms and
definition
• A line extends in one
  dimension. It is                         l
  usually represented
  by a straight line with                 A

  two arrowheads to
  indicate that the line
  extends without end
                            B
  in two directions. In         Line   l or AB
  this book, lines are
  always straight lines.
Using Undefined terms and
definition
A plane extends in two
  dimensions. It is usually
  represented by a shape
  that looks like a tabletop          A
  or wall. You must
  imagine that the plane
                                            C
  extends without end even       B
  though the drawing of a
  plane appears to have
  edges.
                               Plane M or plane ABC
A few basic concepts . . .
 Must be commonly understood without
  being defined. One such concept is the
  idea that a point lies on a line or a plane.
 Collinear points are points that lie on the
  same line.
 Coplanar points are points that lie on the
  same plane.
Ex. 1: Naming Collinear and
Coplanar Points
a.   Name three points
                                  H
     that are collinear
                              G


Solution:                 D       E

D, E and F lie on the
    same line, so they
    are collinear.
Ex. 1: Naming Collinear and
Coplanar Points
b.   Name four points that
     are coplanar.                     H

                                   G

Solution:
                               D       E
D, E, F, and G lie on the
     same plane, so they are
     coplanar. Also D, E, F,
     and H are coplanar;
     although, the plane
     containing them is not
     drawn.
Ex. 1: Naming Collinear and
Coplanar Points
c.   Name three points
     that are not                 H

     collinear.               G


                                  E
Solution:                 D

There are many correct
    answers. For
    instance, points H,
    E, and G do not lie
    on the same line.
More . . .
   Another undefined
    concept in geometry is                  l
    the idea that a point on a
    line is between two other
    points on the line. You
    can use this idea to
    define other important
    terms in geometry.
   Consider the line AB
                                 Line   l or AB
    (symbolized by AB).
More . . .                 A                        B

   The line segment or        Segment AB

    segment AB                                          l
    (symbolized by AB)                        B
    consists of the
    endpoints A and B,
    and all points on AB
    that are between A          A
    and B.                                  Line   l or AB
More . . .                    A            B


   The ray AB                    Ray AB

    (symbolized by AB)                                    l
    consists of the initial                      B
    point A and all points
    on AB that lie on the
    same side of A as
    point B.                       A
                                               Line   l or AB
More . . .                      A                    B


   Note that AB is the    Ray BA
    same as BA and AB is                       l
    the same as BA.                   B
    However, AB and BA
    are not the same.
    They have different
    initial points and      A
    extend in different             Line   l or AB
    directions.
    More . . .
 If C is between A and B,
  then CA and CB are
                                                  l
  opposite rays.
 Like points, segments and              B
  rays are collinear if they       C
  lie on the same line. So,
  any two opposite rays are
  collinear. Segments, rays    A
  and lines are coplanar if            Line   l or AB
  they lie on the same
  plane.
Ex. 2: Drawing lines, segments
and rays
   Draw three noncollinear points J, K, and L.
    Then draw JK, KL and LJ.
                                  Draw J, K and L
                    K
                                  Then draw JK




      J
                             L
Ex. 2: Drawing lines, segments
and rays
   Draw three noncollinear points J, K, and L.
    Then draw JK, KL and LJ.

                    K
                                  Draw KL



     J
                             L
Ex. 2: Drawing lines, segments
and rays
   Draw three noncollinear points J, K, and L.
    Then draw JK, KL and LJ.

                    K
                                  Draw LJ



     J
                             L
Ex. 3: Drawing Opposite Rays
   Draw two lines. Label
    points on the lines and
    name two pairs of           M
    opposite rays.                      Q


Solution: Points M, N, and          X
    X are collinear and X is
    between M and N. So XM
    and XN are opposite rays.   P       N
Ex. 3: Drawing Opposite Rays
   Draw two lines. Label
    points on the lines and
    name two pairs of           M
    opposite rays.                      Q


Solution: Points P, Q, and X        X
    are collinear and X is
    between P and Q. So XP
    and XQ are opposite rays.   P       N
Goal 2: Sketching Intersections of
Lines and Planes
 Two or more geometric intersect if they
  have one or more points in common. The
  intersection of the figures is the set of
  points the figures have in common.
 Activity: p. 12 – Modeling intersections.
    – Use two index cards. Label them as shown
      and cut slots along each card.
       Complete the exercise and place the completed
        questions in your lab section labeling this Lab 1.2.
Ex. 4: Sketching intersections
    Sketch the figure
     described.
a.   A line that intersects a
     plane in one point
    Draw a plane and a
     line.
    Emphasize the point
     where they meet.
    Dashes indicate where
     the line is hidden by
     the plane
Ex. 4: Sketching intersections
    Sketch the figure
     described.
b.   Two planes that
     intersect in a line
    Draw two planes.
    Emphasize the line
     where they meet.
    Dashes indicate where
     one plane is hidden by
     the other plane.
                     Union
   An angle is the union ( ) of two
    noncollinear rays with a common
    endpoint.
     In geometric notation:

            AB     AC  CAB

                              B        Union means
              A                        combining two
                                       sets into a
                                       larger set
                          C
                Intersection
   The intersection symbol ( ) means the
    objects that figures have in common.

    For example:   ABC    BC  BC

                          Note: Segment BC is
           A              common with itself
                          and the triangle.


            B                          C
Now for some practice…

Assignment . .
 Pages   7-8 # 1-13 all

				
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