basic euclidean geometry
DESCRIPTION
Basic Euclidean Geometry. As we begin our study of geometry, keep in mind that these basic ideas haven’t changed in over 3000 years! All of what we know about “Earth Measure” can be explained by these ideas: What is a location / address? What is a collection of locations? - PowerPoint PPT PresentationTRANSCRIPT
BASIC EUCLIDEAN GEOMETRY
As we begin our study of geometry, keep in mind that these basic ideas haven’t changed in over 3000 years!
All of what we know about “Earth Measure” can be explained by these ideas:
What is a location / address?
What is a collection of locations?
What is a collection of collections?
Using Undefined terms and definition
• A point has no dimension. It is usually represented by a small dot.
A
Point A
Always name a point with a single capital letter…
Using Undefined terms and definition• 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
You can name a line using any two letters on the line or a small scripted letter
Using Undefined terms and definitionA 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 ABCYou can name a plane using 3 non-collinear points or a capital scripted letter
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 Pointsa. Name three points
that are collinear
Solution:
D, E and F lie on the same line, so they are collinear.
G
D E F
H
Ex. 1: Naming Collinear and Coplanar Pointsb. Name four points that
are coplanar.
Solution:
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.
G
D E F
H
Ex. 1: Naming Collinear and Coplanar Pointsc. Name three points
that are not collinear.
Solution: There are many correct
answers. For instance, points H, E, and G do not lie on the same line.
G
D E F
H
More . . .
• Another undefined concept in geometry is 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 (symbolized by AB).
l
Line l or AB
More . . .
• The line segment or segment AB (symbolized by AB) consists of the endpoints A and B, and all points on AB that are between A and B.
l
Line l or AB
A
A
B
B
Segment AB
More . . .
• The ray AB (symbolized by AB) consists of the initial point A and all points
on AB that lie on the same side of A as point B.
l
Line l or AB
A
A
B
B
Ray AB
More . . .
• Note that AB is the same as BA and AB is the same as BA. However, AB and BA are not the same. They have different initial points and extend in different directions.
l
Line l or AB
A
A
B
B
Ray BA
More . . . • If C is between A and B, then CA and CB are opposite rays.
• Like points, segments and rays are collinear if they lie on the same line. So, any two opposite rays are collinear. Segments, rays and lines are coplanar if they lie on the same plane.
l
Line l or AB
A
C
B
Ex. 2: Drawing lines, segments and rays• Draw three noncollinear points J, K, and L. Then draw
JK, KL and LJ.
J
K
L
Draw J, K and L
Then draw JK
Ex. 2: Drawing lines, segments and rays• Draw three noncollinear points J, K, and L. Then draw
JK, KL and LJ.
J
K
L
Draw KL
Ex. 2: Drawing lines, segments and rays• Draw three noncollinear points J, K, and L. Then draw
JK, KL and LJ.
J
K
L
Draw LJ
Ex. 3: Drawing Opposite Rays
• Draw two lines. Label points on the lines and name two pairs of opposite rays.
Solution: Points M, N, and
X are collinear and X is
between M and N. So XM
and XN are opposite rays. P
MQ
N
X
Ex. 3: Drawing Opposite Rays
• Draw two lines. Label points on the lines and name two pairs of opposite rays.
Solution: Points P, Q, and
X are collinear and X is
between P and Q. So XP
and XQ are opposite rays. P
MQ
N
X
SOLUTION
EXAMPLE 1 Name points, lines, and planes
b. Name three points that are collinear.Name four points that are coplanar.
a. Other names for PQ are QP and line n. Other names for plane R are plane SVT and plane PTV.
a. Give two other names for PQ and for plane R.
EXAMPLE 1 Name points, lines, and planes
b. Points S, P, and T lie on the same line, so they are collinear. Points S, P, T,and V lie in the same plane, so they are coplanar.
GUIDED PRACTICE for Example 1
ANSWER TS, PT; point V
1. Use the diagram in Example 1. Give two other names for ST . Name a point that is not coplanar with points Q, S, and T.
EXAMPLE 2 Name segments, rays, and opposite rays
b. Name all rays with endpoint J . Which of these rays are opposite rays?
SOLUTION
a. Another name for GH is HG .
b. The rays with endpoint J are JE , JG , JF , and JH . The pairs of opposite rays with endpoint J are JE and JF , and JG and JH .
a. Give another name for GH .
GUIDED PRACTICE for Example 2
3. Are HJ and JH the same ray ? Are HJ and HG the same ray? Explain.
ANSWER FE
ANSWER
No; the rays have different endpoints.Yes; points J and G lie on the same side of H.
2. Give another name for EF