1. Basic Notions in Geometry
TC2MA234
Prepared by Ms. Elizabeth Ann Vasu

2. Geometry
The word geometry comes from Greek words meaning “to measure the Earth”
Basically, Geometry is the study of shapes and is one of the oldest branches of mathematics
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3. The Greeks and Euclid
Our modern understanding of geometry began with the Greeks over 2000 years ago.
The Greeks felt the need to go beyond merely knowing certain facts to being able to prove why they were true.
Around 350 B.C., Euclid of Alexandria wrote The Elements, in which he recorded systematically all that was known about Geometry at that time.
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4. The Elements
Knowing that you can’t define everything and that you can’t prove everything, Euclid began by stating three undefined terms:
Point
(Straight) Line
Plane (Surface)
is that which has no part
is a line that lies evenly with the points on itself
is a plane that lies evenly with the straight lines on itself
Actually, Euclid did attempt to define these basic terms . . .
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5. Point
A point is a location in a plane or space. It is labeled point K. K
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6. Line
A line extends forever in both directions. A line is labeled with two arrows on the ends and written as AB. B A
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7. Plane A Plane is a flat surface of any shape that can go on forever.

8. Ray
A ray has one endpoint and the other is like a line, extending indefinitely. This means one point and one arrow. A ray is labeled as DC. D C

9. Line Segment
A line segment has two endpoints, making it a definite length. It can be labeled as GH. G H

10. Vocabulary review: Identify the figure
Plane: 2-dimensional space
Line: Straight line that extends forever in both directions.
Point: A “spot” on a line
Ray: Starts at one point & extends forever in ONE direction
Line Segment: Part of a line that is between two points
Have students identify the figure
point •
line
ray
The Plane
line segment

11. Naming Rays Lines and Segments
Line Segments:

12. Match the term with the figure
Ray B. Line C. Line Segment
1) 2) 3) P R Q S B A B C A

13. Exercise 1 What is another way to name line ‘l ‘ ?
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14. VERTICAL LINE A VERTICAL LINE goes up & down
The candy bars are vertical

15. HORIZONTAL LINE A HORIZONTAL LINE goes “across” (left and right)
The candy bars are Horizontal

16. Opposite Rays
If a point B lies on AC between A and B, then BA and BC are opposite rays
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17. Collinear Points
Collinear points are points on the same line. When, A, B and C are three collinear points, then B is between A and C if AB + BC = AC. Point D is not between A and C. D
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18. Exercise If AB = 27 and BC = 13, find AC.
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19. Exercise:
If RE = 4x + 7, ET = 2(3x – 4) and RT = 43, then find the value of x, RE and ET.
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20. Coplanar Points
Coplanar points are points on the same plane. Points O, P, Q are coplanar points.
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21. Exercise 2 Name three collinear points. Name four coplanar points
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22. Congruent Lines Segments
Line segments that have the same length are called congruent line segments. AB = EF ( lengths are equal) AB ≈ EF ( is congruent to)
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23. Exercise If AB ≈ CD, AB = 2x – 5 and CD = 7x – 55, find AB.
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24. Exercise Given that ABCD is a line with AB ≈ CD, prove that AC ≈ BD
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25. Non-coplanar Points
Non-coplanar points are points that cannot all be placed on a single plane. Points O, P, Q and R are non-coplanar points.
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26. Coplanar Lines
Lines in the same plane are called coplanar lines. Lines OQ, QR and OR are coplanar. Lines OQ and PQ are coplanar.
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27. Intersecting lines
They are line with exactly one point in common. Lines AB and CD intersect at point E.
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28. Skew Lines
Lines for which there is no plane that contains them (they are not coplanar lines) Lines PQ and OR are skew lines.
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29. Concurrent lines
Three or more lines that intersect at the same point. Lines PQ, OQ and RQ are concurrent lines
A set of concurrent lines.
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30. Parallel lines
Two coplanar lines are parallel if they do not intersect, or if they are the same line. A line is parallel to itself. Line m is parallel to n, written m||n; also m||m and n||n m n
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31. Exercise Name the intersection of PQ and line k.
Name the intersection of plane A and plane B.
Name the intersection of line k and plane A.
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32. Exercise: Sketch a plane and a line that is in the plane.
Sketch a plane and a line that does not intersect the plane.
Sketch a plane and a line that intersects the plane.
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33. Axioms and Theorems
Euclid considered geometric statements concerning points, lines, circles and planes. Some statements he called axioms/postulates, statements that could not be proven. Others that could be proven he called theorems.
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34. Axioms
There is exactly one line that contains any two distinct points.
If two points lie on a plane, then the line containing the points lie in the plane.
If two distinct planes intersect, then their intersection is a line.
There is exactly one plane that contains any three distinct non-collinear points
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35. Axioms Through any two points one and only one line can be drawn.
Two lines cannot intersect in more than one point.
The shortest distance between any two points is the line segment joining them.
A line segment has only one mid-point.
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36. Exercise:
If two distinct lines AEB and CED intersect at a point F, what must be true about point E and F? Use the above axioms to justify your answer.
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37. Exercise: If CE = CF, CD = 2CE and CB = 2CF, then prove that CD = CB.
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38. Theorems A line and a point not on the line determine a unique plane.
Two distinct parallel lines determine a unique plane.
Two distinct intersecting lines determine a unique plane.
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