1. Points, Lines, and Planes
Lesson 1.3
Points, Lines, and Planes

2. Objectives: Identify and model points, lines, and planes.
Use postulates and undefined terms.

3. Key Vocabulary Undefined term Point Line Plane Postulate Collinear
Coplanar
Segment
Ray
Endpoint
Space

4. Postulates Postulate 1: Two Points Determine a Line.
Postulate 2: Three Points Determine a Plane.

5. What is a definition?
A definition uses known words to describe a new word.
Undefined terms – not formally defined
There are 3 undefined terms in geometry.
Point Line Plane

6. Basic Geometric Objects
The 3 Undefined Terms of Geometry represent these three objects…
Points
Lines
Planes
Every object in geometry has a
Definition (except the 3 undefined terms) and is defined using the 3 undefined terms.
Symbol or Name
Image or Diagram

7. Points Points do not have actual size. How to Sketch: Using dots
How to label:
Use capital letters
Never name two points with the same letter
(in the same sketch). A B C A

8. Points Continued
In Algebra, points are represented by ordered pairs of numbers (x, y).
Points make up every algebraic curve.
For example, the collection of points that satisfy the following equations create the curves shown below each.

9. Example: Which of the following is the best representation of a point?
A clock
A floor tile
The buttons on the projector.

10. Lines
Lines extend indefinitely and have no thickness or width.
How to sketch : using arrows at both ends.
How to name: 2 ways
(1) small script letter – line n
(2) any two points on the line -
Never name a line using three points - n A B C

11. Lines Continued How many points are on a line?
Infinitely Many!
More importantly…how many points determine a line?
Exactly two.
(Therefore) another way to name a line is by using two points on the line.
Diagram of Line
Name of Line A B or

12. Lines (Continued)
So, to name a line by using two points, we write the two capital letters of the points next to each other and then draw a little line symbol over them.
It doesn’t matter which letter comes first.
Important: the line symbol above the letters has arrows on both ends.

13. Example: In this room the best representation of a line is:
The place where the wall and the ceiling meet.
The floor.
The power cord on the projector.

14. Collinear Points
Collinear points are points that lie on the same line. (The line does not have to be visible.)
A point lies on the line if the coordinates of the point satisfy the equation of the line.
Ex: To find if A (1, 0) is collinear with
the points on the line y = -3x + 3.
Substitute x = 1 and y = 0 in the equation.
0 = -3 (1) + 3
0 =
0 = 0
The point A satisfies the equation, therefore the point is collinear
with the points on the line. A B C
Collinear C A B
Non collinear

15. Plane

16. Plane
NOT!

17. Planes
A plane is a flat surface that extends indefinitely in all directions.
How to sketch: Use a parallelogram (four sided figure)
How to name: 2 ways
(1) Capital script letter – Plane M
(2) Any 3 non collinear points in the plane - Plane: ABC/ ACB / BAC / BCA / CAB / CBA A M B C
Horizontal Plane
Vertical Plane
Other

18. Example:
True or False?
The ceiling is a real-life example of a plane.

19. Example: Different planes in a figure:B
Plane ABC
Plane EFG
Plane BCG
Plane ADH
Plane ABF
Plane CDH
Etc. D C E F H G

20. Example: Other planes in the same figure:
Any three non collinear points determine a plane!
Plane AFG
Plane ACG
Plane ACH
Plane AGB
Plane BDG
Etc.

21. Example 1 A. Use the figure to name a line containing point K.
Answer: The line can be named as line a.
There are three points on the line. Any two of the points can be used to name the line.

22. Example 1 B. Use the figure to name a plane containing point L.
You can also use the letters of any three noncollinear points to name the plane. plane JKM plane KLM plane JLM
Answer: The plane can be named as plane B.

23. Example 1
The letters of each of these names can be reordered to create other acceptable names for this plane. For example, JKM can also be written as JMK, MKJ, KJM, KMJ, and MJK. There are 15 different three-letter names for this plane.

24. Your Turn A. Use the figure to name a line containing the point X.
A. line X
B. line c
C. line Z D.

25. Your Turn B. Use the figure to name a plane containing point Z.
A. plane XY
B. plane c
C. plane XQY
D. plane P

26. Postulates
In geometry, a postulate is a statement that describes a fundamental relationship between the basic terms of geometry.
Postulates are always accepted as true without proof.
Axiom is another name for postulate. A statement that is accepted as true.

27. Postulates
The basic concepts about points, lines, and planes can be stated as postulates.
Postulate 1 – Through any two points, there is exactly one line.
Postulate 2 – Through any three points not on a line there is exactly one plane. A C A B C

28. Example 2: Use the diagram at the right. Name three points.
Name two lines.
Name two planes. a. b. c.
SOLUTION
D, E, and F are points. a.
Line m and line p b.
Q and R are planes. c. 28

29. Your Turn: Use the diagram at the right. 1. Name two lines. ANSWER
Sample answer: line m and line n 2.
Name two planes.
ANSWER
plane S and plane T 3.
Name three points that are collinear.
ANSWER
C, D, and E

30. Example: Coplanar Objects
Coplanar objects (points, lines, etc.) are objects that lie on the same plane. The plane does not have to be visible.
Are the following points coplanar?
A, B, C ?
Yes
A, B, C, F ? No
H, G, F, E ?
Yes
E, H, C, B ?
Yes
A, G, F ?
Yes
C, B, F, H ? No

31. Example 3: Use the diagram at the right.
Name three points that are collinear.
Name four points that are coplanar.
Name three points that are not collinear. a. b. c.
SOLUTION
Points D, E, and F lie on the same line. So, they are collinear. a.
Points D, E, F, and G lie on the same plane, so they are coplanar. b.
Points H, E, and G do not lie on the same line. There are many other correct answers. c. 31

32. Your Turn: Use the diagram at the right. 1.
Name three points that are not collinear.
ANSWER
Sample answer: B, C, and D 2.
Name four points that are coplanar.
ANSWER
B, C, D, and E 3.
Name two lines that are coplanar.
ANSWER
p and n, or p and m

33. Line Segment Definition:
Part of a line that consists of two points called the endpoints and all points between them.
How to sketch:
How to name:
AB (without a symbol) means the length of the segment or the distance between points A and B.

34. Line Segment Line Line Segment
When naming a line segment we draw a picture of a line segment (without arrows or dots on the ends) above the two capital letters denoting the endpoints.
Note well the difference between the name of a line and the name of a line segment:
Line
Line Segment

35. Rays
A ray
is part of a line which has one endpoint and extends infinitely in one direction.
named stating the endpoint first and then any other point on the ray. A B
Ray AB

36. We could label this ray as AB, AC, or AD but not CA.
Labeling Rays
We could label this ray as AB, AC, or AD but not CA. D C B A

37. More about Rays
If you choose a point on a line, that point determines exactly two rays called 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. P Q R
QP and QR are opposite rays.

38. Example 4:Draw Lines, Segments, and Rays
Draw three noncollinear points, J, K, and L. Then draw JK, KL, and LJ.
SOLUTION
1. Draw J, K, and L.
2. Draw JK.
3. Draw KL.
4. Draw LJ. 38

39. Your Turn: 1. Draw four points as shown. 2.
ANSWER 2.
Draw the lines AB and AC. Are the lines the same? Explain.
ANSWER
yes, since A, B, and C, are collinear points.

40. Your Turn: 3.
Draw the line segments AC and BD. Are the segments the same? Explain.
ANSWER
no, since the segments have different endpoints. 4.
Draw the rays CA and CB. Are the rays the same? Explain.
ANSWER
yes, since both rays have endpoint C and extend in the same direction.

41. What is space?
The set of all points.

42. Joke Time How do you know when it’s raining cats and dogs?
When you step in a poodle!
Why did the apple go out with a fig?
Because it couldn’t find a date.

43. Assignment
Section 1.3, pg. 17 – 20: #1 – 53 odd, 63, 65