This document provides an introduction to basic geometry concepts including: - Axiomatic systems and undefined terms like points, lines, and planes. - Linear and planar notions such as collinear points, intersecting lines, and parallel planes. - Properties of points, lines, and planes including that three noncollinear points determine a unique plane. - Angles are formed by two rays with a common endpoint, and can be measured in degrees, minutes, and seconds. Different types of angles are defined including right, acute, and obtuse angles. - Perpendicular lines intersect to form right angles, and a line is perpendicular to a plane if it is perpendicular to every line in the plane

1. Math 310
Section 9.1
Geometry Introduction

2. Axiomatic System
A logical system which possesses an explicitly
stated set of axioms from which theorems can
be derived.
(mathworld.wolfram.com)

3. www.xkcd.com

4. Undefined Terms
 Point
 Line
 Plane

5. “One must be able to say at all
times—instead of points, straight
lines, and planes—tables, chairs,
and beer mugs.”
- David Hilbert

6. Linear “Notions”
 Collinear points
 Is between
 Line segment (segment)
 Ray

7. Collinear points (& non collinear)
B C
D
E

8. Is Between
B C
E

9. Line Segment
F
G

10. Ray
H
I

11. Planar “Notions”
 Coplanar points
 Noncoplanar points
 Coplanar lines
 Skew lines
 Intersecting lines
 Concurrent lines
 Parallel lines

12. Coplanar & Noncoplanar Points
A C
B
D

13. Coplanar lines
F
I
H
G
E
D

14. Skew Lines
F
I
H
E

15. Intersecting Lines
K
L
J

16. Concurrent Lines
K
L
J
M

17. Parallel Lines
N O
Q
P

18. Properties of “tables, chairs and beer
mugs”
1. There is exactly one line that contains any two
distinct points
2. If two points lie in a plane, then the line
containing the points lies in the plane.
3. If two distinct planes intersect, then their
intersection is a line.
4. There is exactly one plane that contains any
three distinct noncollinear points.

19. Properties (cont)
5. A line and a point not on the line determine a
plane.
6. Two parallel lines determine a plane
7. Two intersecting lines determine a plane.

20. Property 1
A B
A
B

21. Property 2

22. Property 3
A
B

23. Property 4

24. Property 5

25. Property 6

26. Property 7

27. Intersecting Planes
 Parallel
 Along a line

28. Parallel Planes

29. Planes Intersecting along a line

30. Angle, Vertex, Side
Def
When two rays share an endpoint, an angle is formed.
The common initial point of the rays is the vertex of
the angle. Each ray is called a side of the angle.

31. Ex.
B
A
C
Vertex
Side
Angle: <ABC

32. Ex.
F
E
G
B
E
F
I
H
J
<EBF
<I
<GFE

33. J K
L
M
N
O
Ex.
Name all six
angles.
<MJK
<NKL
<OLJ
<KJL
<LKJ
<JLK

34. Angle Measure
To measure an angle we use the unit degree. It
measures the “opening” of the angle. The
largest angle measure is 360° and the smallest is
0°. A complete rotation about a point is 360°.
For more accuracy, angles can be further
measure in minutes, and seconds. Each degree
is divided into 60 minutes, and each minute is
divided into 60 seconds.

35. Ex.
Add: 45°23’47” and 62°36’51”
45°23’47” + 62°36’51” = 108°0’38” or 108°38”
Add: 145°17’4” and 220°31’32”
145°17’4” + 220°31’32” = 365°48’36” = 5°48’36”

36. Ex.
Solve for x.
B
C
A D
m<ABC = 80°
m<ABD = 30°
m<DBC = (x – 25)°
m<ABC = 82°
m<ABD = (x – 13)°
m<DBC = (x + 7)°
x = 75
x = 44

37. Protractor
A protractor is a standard tool for measuring
angles. To use, line the vertex up with the
center of the base of the protractor and line one
side of the angle up with the 0° mark. Now
measure from 0°, increasing, until you see the
other side of the angle and read the mark.

38. Types of Angles
 Obtuse
 Acute
 Right
 Straight

39. Ex.
I
H
J
B
E
F
B
A
C
acute
straight
obtuse
F
H
G
right

40. Perpendicular Lines
Def.
Two lines are perpendicular if they intersect and
form right angles.
perpendicular not
perpendicular

41. Line Perpendicular to a Plane
A line is perpendicular to a plane if it is
perpendicular to every line, contained in the
plane, passing through the point of intersection.

42. Ex.

43. Ex.

44. Questions
 Is it possible for a line intersecting a plane to be
perpendicular to exactly one line in the plane
through its intersection with the plane?
 Can a line intersecting a plane be perpendicular
to exactly two distinct lines in the plane going
through the point of intersection?
 Yes
 No

45. Line Perpendicular to a Plane
Thrm
A line perpendicular to two distinct lines in the
plane through its intersection with the plane is
perpendicular to the plane.