1. Geometry
3.1 Line & Angle Pairs

2. Geometry 3.1 Lines and Angles
Goals
Identify relationships between lines.
Identify angles formed by lines.
Identify lines and planes
January 13, 2019
Geometry 3.1 Lines and Angles

3. Geometry 3.1 Lines and Angles
Coplanar: Lie on
The same plane
Parallel Lines
Coplanar lines that do not intersect.
m || n m n
January 13, 2019
Geometry 3.1 Lines and Angles

4. Geometry 3.1 Lines and Angles
Skew Lines
Lines that do not intersect and are not coplanar. s r
January 13, 2019
Geometry 3.1 Lines and Angles

5. Geometry 3.1 Lines and Angles
Parallel Planes
Planes that don’t intersect.
January 13, 2019
Geometry 3.1 Lines and Angles

6. Segments and Rays can be parallel.
January 13, 2019
Geometry 3.1 Lines and Angles

7. Geometry 3.1 Lines and Angles
Visualization B D
Parallel E A
Skew G
Perpendicular F
Think of a rectangular box.
January 13, 2019
Geometry 3.1 Lines and Angles

8. Geometry 3.1 Lines and Angles
History
Lesson
January 13, 2019
Geometry 3.1 Lines and Angles

9. Geometry 3.1 Lines and Angles
Euclid
Lived about 325 BC to 265 BC.
Worked in Alexandria, Egypt.
Wrote the first geometry book, The Elements, which is still in print.
The Elements begins with five postulates.
January 13, 2019
Geometry 3.1 Lines and Angles

10. Geometry 3.1 Lines and Angles
Euclid’s 5th Postulate
Given a line segment that crosses two lines in a way that the sum of the inner angles on the same side is less than two right angles, then the two lines will eventually meet (on that side of the line segment).
January 13, 2019
Geometry 3.1 Lines and Angles

11. Geometry 3.1 Lines and Angles
Modern Version
The 5th Postulate was modernized and simplified by John Playfair (1748 – 1819).
Although it reads like a theorem it isn’t, and despite 200 years of attempts, it cannot be proved. It is a postulate.
Attempts to prove it led to some remarkable new ideas that will be seen momentarily.
January 13, 2019
Geometry 3.1 Lines and Angles

12. Postulate 3.1: Parallel Postulate
If there is a line and a point not on the line, then there is exactly one line through the point parallel to the given line.
January 13, 2019
Geometry 3.1 Lines and Angles

13. Postulate 3.2: Perpendicular Postulate
If there is a line and a point not on the line, then there is exactly one line through the point perpendicular to the given line.
January 13, 2019
Geometry 3.1 Lines and Angles

14. Geometry 3.1 Lines and Angles
Transversals
A transversal is a line that intersects two or more coplanar lines at different points. t m n
Transversal
January 13, 2019
Geometry 3.1 Lines and Angles

15. This is not a transversal.
The lines intersect at only one point.
January 13, 2019
Geometry 3.1 Lines and Angles

16. Geometry 3.1 Lines and Angles
Special Angle Pairs
These are identified by their positions relative to one another. Learn to identify them and name them. These are listed on page 128 of the text. 1 2 4 3 5 6 8 7
January 13, 2019
Geometry 3.1 Lines and Angles

17. Geometry 3.1 Lines and Angles
Corresponding angles
1 & 5
2 & 6
3 & 7
4 &  8 1 2 4 3 5 6 8 7
January 13, 2019
Geometry 3.1 Lines and Angles

18. Alternate Exterior Angles
1 & 7
2 & 8 1 2 4 3 5 6 8 7
January 13, 2019
Geometry 3.1 Lines and Angles

19. Alternate Interior Angles
4 & 6
3 & 5 1 2 4 3 5 6 8 7
January 13, 2019
Geometry 3.1 Lines and Angles

20. Same Side Interior Angles
Your book calls these Consecutive Interior Angles. We will also use Same Side Interior.
4 & 5
3 & 6 1 2 4 3 5 6 8 7
January 13, 2019
Geometry 3.1 Lines and Angles

21. Geometry 3.1 Lines and Angles
Abbreviations
Corresponding Angles
Corr s
Alternate Exterior Angles
Alt Ext s
Alternate Interior Angles
Alt Int s
Same Side Interior Angles
SS int s
January 13, 2019
Geometry 3.1 Lines and Angles

22. More on the Parallel Postulate
Euclid:
Given a line segment that crosses two lines in a way that the sum of the inner angles on the same side is less than two right angles, then the two lines will eventually meet (on that side of the line segment).
What if this isn’t true?
January 13, 2019
Geometry 3.1 Lines and Angles

23. Geometry 3.1 Lines and Angles
Is it possible that m || n || l? l n m
January 13, 2019
Geometry 3.1 Lines and Angles

24. If we assume it is possible…
Bernard Riemann (1826 – 1866)
Hyperbolic Geometry
January 13, 2019
Geometry 3.1 Lines and Angles

25. Geometry 3.1 Lines and Angles
Euclidean Space
January 13, 2019
Geometry 3.1 Lines and Angles

26. Geometry 3.1 Lines and Angles
Hyperbolic Space
January 13, 2019
Geometry 3.1 Lines and Angles

27. Which one was the correct view?
Albert Einstein (1879 –1955)
On the small scale, Euclid is all that is needed.
On the large scale of the universe, Riemann is correct.
This was contained in his brilliant theory, Special Relativity.
January 13, 2019
Geometry 3.1 Lines and Angles

28. Geometry 3.1 Lines and Angles
A simple postulate by Euclid, has led to one of the most profound theories in understanding the universe:
The universe is not flat, i.e. it is not Euclidean.
January 13, 2019
Geometry 3.1 Lines and Angles

29. Geometry 3.1 Lines and Angles
Space is curved.
January 13, 2019
Geometry 3.1 Lines and Angles

30. Geometry 3.1 Lines and Angles
But not for us.
In our study of geometry, we assume the Parallel Postulate to be true.
January 13, 2019
Geometry 3.1 Lines and Angles

31. Geometry 3.1 Lines and Angles
Summarize
January 13, 2019
Geometry 3.1 Lines and Angles