1. 1.1: Identify Points, Lines, & Planes 1.2: Use Segments & Congruence
02 Building Blocks of Geometry
1.1 and Building Blocks of Geometry
1.1: Identify Points, Lines, & Planes 1.2: Use Segments & Congruence
Objectives:
To learn the terminology and notation of the basic building blocks of geometry
To use the Ruler and Segment Addition Postulates
To construct congruent segments with compass and straightedge 1

2. Vocabulary Point Line Segment Line Ray Plane
In your notes, define each of these without your book. Draw a picture for each word and leave a bit of space for additions and revisions.
Point
Line Segment
Line
Ray
Plane

3. Undefined Terms?
In geometry, we always try to define things in simpler terms. Point, line, and plane are considered undefined terms, however, and cannot be made any simpler, so we just describe them.
What the Ancient Greeks said:
“A point is that which has no part. A line is breadthless length.”

4. Undefined Terms?
In geometry, we always try to define things in simpler terms. Point, line, and plane are considered undefined terms, however, and cannot be made any simpler, so we just describe them.
What the Ancient Chinese said:
“The line is divided into parts, and that part which has no remaining part is a point.”

5. Mathematical model of a point
Points
Basic unit of geometry
No size, only location
ZERO dimensions
Represented by a dot and named by a capital letter
Mathematical model of a point

6. Points Mathematical model of a point
A star is a physical model of a point

7. Mathematical model of a line
Lines
Straight arrangement of points
No width, only length
Extends forever in 2 directions
ONE dimension
Named by two points on the line: line AB or BA or or
It can be named with a lower-case script letter: l l
Mathematical model of a line

8. Lines Mathematical model of a line
Spaghetti is a physical model of a line

9. Mathematical model of a line
Lines
How many lines can you draw through any two points?
ONE
Mathematical model of a line

10. Collinear Points Collinear points are points that --?--.
Points A, B, and C are collinear
Lie on a line
What do you supposed NON-collinear means?

11. Mathematical model of a plane
Planes
Flat surface that extends forever
Length and width but no height (2-D)
Represented by a 4-sided figure and named by a capital script letter or 3 (non-collinear) letters on the same plane.
Mathematical model of a plane

12. Planes Mathematical model of a plane
Flattened dough is a physical model of a plane

13. Mathematical model of a plane
Planes
How many points does it take to define a plane?
(tell where the plane is in space?)
THREE
Mathematical model of a plane

14. Coplanar Points Coplanar points are points that --?--.
Points A, B, and C are coplanar
Lie on the same plane
What do you suppose NON-coplanar means?

15. Hierarchy of Building Blocks
Space
3-D
Planes
2-D
Lines
1-D
Points
Space is the set of all points
0-D

16. A Romance of Many Dimensions
Are there more than three spatial dimensions? ?
Point
Segment
Square
Cube
0-D
1-D (Length)
2-D (Area)
3-D (Volume)
1 point
2 points
4 points
8 points
0 “sides”
2 “sides”
4 “sides”
6 “sides”

17. A Romance of Many Dimensions
Are there more than three spatial dimensions?
Point
Segment
Square
Cube
Hypercube
0-D
1-D (Length)
2-D (Area)
3-D (Volume)
1 point
2 points
4 points
8 points
0 “sides”
2 “sides”
4 “sides”
6 “sides”

18. A Romance of Many Dimensions
Are there more than three spatial dimensions?
Point
Segment
Square
Cube
Hypercube
0-D
1-D (Length)
2-D (Area)
3-D (Volume)
4-D (Hypervolume)
1 point
2 points
4 points
8 points
0 “sides”
2 “sides”
4 “sides”
6 “sides”

19. A Romance of Many Dimensions
Are there more than three spatial dimensions?
Point
Segment
Square
Cube
Hypercube
0-D
1-D (Length)
2-D (Area)
3-D (Volume)
4-D (Hypervolume)
1 point
2 points
4 points
8 points
16 points
0 “sides”
2 “sides”
4 “sides”
6 “sides”

20. A Romance of Many Dimensions
Are there more than three spatial dimensions?
Point
Segment
Square
Cube
Hypercube
0-D
1-D (Length)
2-D (Area)
3-D (Volume)
4-D (Hypervolume)
1 point
2 points
4 points
8 points
16 points
0 “sides”
2 “sides”
4 “sides”
6 “sides”
8 “sides”

21. WATCH the following Computer animation of hypercube
Explanation of 4th dimension

22. Example 1 Give two other names for and plane R.
Name three points that are collinear.
Name four points that are coplanar.

23. Line Segment
A line segment consists of two endpoints and all the collinear points between them.
Line segment AB or
Endpoints

24. Congruent Segments
Congruent segments are line segments that have the same length.
Symbol for congruent

25. Copying a Segment
We’re going to try making two congruent segments using only a compass and a straightedge. Here, we’re not using a ruler to measure the length of the segment!

26. Copying a Segment
Draw segment AB.

27. Copying a Segment
Draw a line with point A’ on one end.

28. Copying a Segment
Put point of compass on A and the pencil on B. Make a small arc.

29. Copying a Segment
Put point of compass on A’ and use the compass setting from Step 3 to make an arc that intersects the line. This is B’.

30. Copying a Segment
Click on the image to watch a video of the construction.
GSP

31. A laser is a physical model of a ray
A ray consists of an endpoint and all of the collinear points to one side of that endpoint.
Ray AB or
A laser is a physical model of a ray

32. Example 2
Ray BA and ray BC are considered opposite rays. Use the picture to explain why.
At what time would the hands of a clock form opposite rays?
3:15, 6:00, 1:35,…

33. Example 3 Give another name for .
Name all rays with endpoint J. Which of these rays are opposite rays? HG JG
JH JF JE

34. Intersection
Two or more geometric figures 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.
The intersection of two planes is a line.
The intersection of two lines is a point.

35. Example 4
What is the length of segment AB? B A

36. Example 4
You basically used the Ruler Postulate to find the length of the segment, where A corresponds to 0 and B corresponds to So AB = |6.5 – 0| = 6.5 cm B A

37. Example 5
Now what is the length of ?
8.5 – 2 = 6.5 A B

38. Ruler Postulate
The points on a line can be matched one to one with the real numbers. The real number that corresponds to a point is its coordinate. The distance between points A and B, written as AB, is the absolute value of the difference of the coordinates of A and B.

39. Example 6
When asked to measure the segment below, Kenny gave the answer 2.7 inches. Explain what is wrong with Kenny’s measurement.
Inches are not divided into tenths

40. Give Them an Inch…
A Standard English ruler has 12 inches. Each inch is divided into parts.
Cut an inch in half, and you’ve got 1/2 an inch.
Cut that in half, and you’ve got 1/4 an inch.
Cut that in half, and you’ve got 1/8 inch.
Cut that in half, and you’ve got 1/16 inch.
Click the ruler and practice measuring both inches and centimeters

41. Example 7
Let’s say you found the length of a segment to be 6’ 7” using your dad’s tape measure. Convert this measurement to the nearest tenth of a centimeter (1” ≈ 2.54 cm).
6 7/12 = 6.58
= 2.54 X
Cross-multiply
X= 16.71

42. Example 8
Use the diagram to find GH. 15

43. Example 8
Use the diagram to find GH. Could you as easily find GH if G was not collinear with F and H? Why or why not?
No, the parts wouldn’t equal the whole. FG + GH = FH

44. Segment Addition Postulate
If B is between A and C, then AB + BC = AC.
If AB + BC = AC, then B is between A and C.
BETWEEN indicates collinear points

45. Example 9
Point A is between S and M. Find x if SA = 2x – 5, AM = 7x + 3, and SM = 25. A M S 25
2x – x + 3
2x – 5 + 7x+3 = 25
9x = 27
X=3

46. Example 10
Point E is between J and R. Find JE given that JE = x2, ER = 2x, and JR = 8.
WORK IT OUT WITH A PICTURE!!
X = 2, why does it not equal -4?

47. Example 11: SAT
Points E, F, and G all lie on line m, with E to the left of F. EF = 10, FG = 8, and EG > FG. What is EG?
Work it out with a picture. 18