1.2 Points, Lines and Planes Geometry Mrs. Spitz Fall 2005.
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1.2 Points, Lines 1.2 Points, Lines and Planes and Planes Geometry Geometry Mrs. Spitz Mrs. Spitz Fall 2005 Fall 2005
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Transcript of 1.2 Points, Lines and Planes Geometry Mrs. Spitz Fall 2005.
1.2 Points, Lines and 1.2 Points, Lines and PlanesPlanes
GeometryGeometry
Mrs. SpitzMrs. Spitz
Fall 2005Fall 2005
Objectives/Assignment:Objectives/Assignment:
Understand and use the basic Understand and use the basic undefined terms and defined terms undefined terms and defined terms of geometry.of geometry.
Sketch the intersections of lines and Sketch the intersections of lines and planes.planes.
Assignment: pp. 13-15 #1-72 allAssignment: pp. 13-15 #1-72 all
Using Undefined terms and Using Undefined terms and definitiondefinitionA 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.
Example what your note page Example what your note page should look like nowshould look like now
Name____________________________
Date___________________Period_____
Class_____________________________
1.2 Points, Lines, & Planes
Objectives:
Assignment:
Using undefined terms and definitions
1.1. Understand and use the basic undefined Understand and use the basic undefined terms and defined terms of geometry.terms and defined terms of geometry.
2.2. Sketch the intersections of lines and Sketch the intersections of lines and planes.planes.
Assignment: pp. 13-15 #1-72 allAssignment: pp. 13-15 #1-72 all
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 Using Undefined terms and definitiondefinition
• A point has no dimension. It is usually represented by a small dot.
A
Point A
Using Undefined terms and Using Undefined terms and definitiondefinition• A line extends in one
dimension. It is usually represented by a straight line with two arrowheads to indicate that the line extends without end in two directions. In this book, lines are always straight lines.
A
B
l
Line l or AB
Using Undefined terms and Using Undefined terms and definitiondefinitionA plane extends in two
dimensions. It is usually represented by a shape that looks like a tabletop or wall. You must imagine that the plane extends without end even though the drawing of a plane appears to have edges.
A
BC
M
Plane M or plane ABC
A few basic concepts . . . A few basic concepts . . .
Must be commonly understood Must be commonly understood without being defined. One such without being defined. One such concept is the idea that a point lies concept is the idea that a point lies on a line or a plane.on a line or a plane.
Collinear pointsCollinear points are points that lie on are points that lie on the same line.the same line.
Coplanar pointsCoplanar points are points that lie on are points that lie on the same plane.the same plane.
Ex. 1: Naming Collinear and Ex. 1: Naming Collinear and Coplanar PointsCoplanar Pointsa.a. Name three Name three
points that are points that are collinearcollinear
Solution: Solution:
D, E and F lie on the D, E and F lie on the same line, so same line, so they are collinear.they are collinear.
G
D E F
H
Ex. 1: Naming Collinear and Ex. 1: Naming Collinear and Coplanar PointsCoplanar Pointsb.b. Name four points Name four points
that are coplanar.that are coplanar.
Solution: Solution:
D, E, F, and G lie on the D, E, F, and G lie on the same plane, so they same plane, so they are coplanar. Also are coplanar. Also D, E, F, and H are D, E, F, and H are coplanar; although, coplanar; although, the plane containing the plane containing them is not drawn.them is not drawn.
G
D E F
H
Ex. 1: Naming Collinear and Ex. 1: Naming Collinear and Coplanar PointsCoplanar Pointsc.c. Name three Name three
points that are points that are not collinear.not collinear.
Solution: Solution: There are many There are many
correct answers. correct answers. For instance, For instance, points H, E, and G points H, E, and G do not lie on the do not lie on the same line.same line.
G
D E F
H
More . . . More . . .
Another undefined Another undefined concept in geometry is concept in geometry is the idea that a point the idea that a point on a line is between on a line is between two other points on two other points on the line. You can use the line. You can use this idea to define this idea to define other important terms other important terms in geometry.in geometry.
Consider the line AB Consider the line AB (symbolized by AB). (symbolized by AB).
l
Line l or AB
More . . . More . . .
The The line segmentline segment or segment AB or segment AB (symbolized by AB) (symbolized by AB) consists of the consists of the endpoints A and B, endpoints A and B, and all points on and all points on AB that are AB that are between A and B.between A and B.
l
Line l or AB
A
A
B
B
Segment AB
More . . . More . . .
The ray AB The ray AB (symbolized by AB) (symbolized by AB) consists of the consists of the initial point A and initial point A and all points on AB all points on AB that lie on the that lie on the same side of A as same side of A as point B.point B.
l
Line l or AB
A
A
B
B
Ray AB
More . . . More . . .
Note that AB is the Note that AB is the same as BA and AB same as BA and AB is the same as BA. is the same as BA. However, AB and However, AB and BA are not the BA are not the same. They have same. They have different initial different initial points and extend points and extend in different in different directions.directions.
l
Line l or AB
A
A
B
B
Ray BA
More . . . More . . . If C is between A and If C is between A and
B, then CA and CB are B, then CA and CB are opposite rays.opposite rays.
Like points, segments Like points, segments and rays are collinear and rays are collinear if they lie on the same if they lie on the same line. So, any two line. So, any two opposite rays are opposite rays are collinear. Segments, collinear. Segments, rays and lines are rays and lines are coplanar if they lie on coplanar if they lie on the same plane.the same plane.
l
Line l or AB
A
C
B
Ex. 2: Drawing lines, segments Ex. 2: Drawing lines, segments and raysand rays
Draw three noncollinear points J, K, Draw three noncollinear points J, K,
and L. Then draw JK, KL and LJ.and L. Then draw JK, KL and LJ.
J
K
L
Draw J, K and L
Then draw JK
Ex. 2: Drawing lines, segments Ex. 2: Drawing lines, segments and raysand rays
Draw three noncollinear points J, K, Draw three noncollinear points J, K,
and L. Then draw JK, KL and LJ.and L. Then draw JK, KL and LJ.
J
K
L
Draw KL
Ex. 2: Drawing lines, segments Ex. 2: Drawing lines, segments and raysand rays
Draw three noncollinear points J, K, Draw three noncollinear points J, K,
and L. Then draw JK, KL and LJ.and L. Then draw JK, KL and LJ.
J
K
L
Draw LJ
Ex. 3: Drawing Opposite Ex. 3: Drawing Opposite RaysRays Draw two lines. Label Draw two lines. Label
points on the lines and points on the lines and name two pairs of name two pairs of opposite rays.opposite rays.
Solution: Points M, N, Solution: Points M, N,
and X are collinear and X are collinear
and X is between M and X is between M
and N. So XM and XN and N. So XM and XN
are opposite rays.are opposite rays.
P
MQ
N
X
Ex. 3: Drawing Opposite Ex. 3: Drawing Opposite RaysRays Draw two lines. Label Draw two lines. Label
points on the lines and points on the lines and name two pairs of name two pairs of opposite rays.opposite rays.
Solution: Points P, Q, Solution: Points P, Q,
and X are collinear and X are collinear
and X is between P and X is between P
and Q. So XP and XQ and Q. So XP and XQ
are opposite rays.are opposite rays.
P
MQ
N
X
Goal 2: Sketching Goal 2: Sketching Intersections of Lines and Intersections of Lines and PlanesPlanes Two or more geometric intersect if they Two or more geometric intersect if they
have one or more points in common. have one or more points in common. The intersection of the figures is the set The intersection of the figures is the set of points the figures have in common.of points the figures have in common.
Activity: p. 12 – Modeling intersections.Activity: p. 12 – Modeling intersections.– Use two index cards. Label them as shown Use two index cards. Label them as shown
and cut slots along each card. and cut slots along each card. Complete the exercise and place the completed Complete the exercise and place the completed
questions in your lab section labeling this Lab questions in your lab section labeling this Lab 1.2.1.2.
Ex. 4: Sketching Ex. 4: Sketching intersectionsintersections Sketch the figure Sketch the figure
described.described.a.a. A line that intersects A line that intersects
a plane in one pointa plane in one point Draw a plane and a Draw a plane and a
line.line. Emphasize the point Emphasize the point
where they meet. where they meet. Dashes indicate Dashes indicate
where the line is where the line is hidden by the planehidden by the plane
Ex. 4: Sketching Ex. 4: Sketching intersectionsintersections Sketch the figure Sketch the figure
described.described.
b.b. Two planes that Two planes that intersect in a lineintersect in a line
Draw two planes.Draw two planes. Emphasize the line Emphasize the line
where they meet.where they meet. Dashes indicate Dashes indicate
where one plane is where one plane is hidden by the other hidden by the other plane.plane.
Coming up soon . . . Coming up soon . . .
Quiz after 1.3Quiz after 1.3 Practice quiz on page 25Practice quiz on page 25
