1. Hon. Geometry Fr Chris Thiel, OFMCap 2016
Parallel Lines
Hon. Geometry
Fr Chris Thiel, OFMCap
2016

2. Definitions Parallel-- coplanar lines that never intersect
Skew--lines that never intersect because they are not coplanar
Definitions

4. Transversal-- A line that goes across
2 or more lines

5. Angles Corresponding Angles Consecutive (Same Side) Interior Angles
Same vs Alternate Side
(of transversal)
Angles
Interior vs Exterior
(of parallel)
Corresponding Angles
Consecutive (Same Side) Interior Angles
Alternate Interior Angles
Alternate Exterior Angles

6. Euclid’s 5th Postulate
If consecutive (same side) interior angles are not supplementary, the lines are not parallel

7. Our Book’s approach
Postulate:
If and only if the lines are parallel,
then Corresponding angles are congruent

8. Theorem: If and only if the lines are parallel,
then Alternate Interior Angles are Congruent

9. Theorem: If and only if the lines are parallel,
then Consecutive (Same Side) Interior Angles are Supplementary

10. Example: #20 p 88

11. Equivalent to Euclid’s Parallel Postulate (John Playfair’s axiom)
Through a point outside a line,
there is exactly one line parallel.

12. Theorem: The Sum of the Interior Angles of a Triangle is 180°

13. Remote Interior Angles Theorem

14. Regular Convex Polygons
Not Regular
Not Convex

15. Induction
2, 4, 6, 8, …
2, 3, 5, 7, 11, …
4, 5, 7, 10, 14, 19, …
1, 4, 9, 16, 25, 36, …
1, 8, 27, 64, 125, 216, …

16. Sum of Interior Angles of Convex Polygons:

17. Interior Angles of Regular Convex Polygons
The supplement (the Exterior Angle) is...
Sum of Exterior Angles of Convex Polygons

18. The Exterior angle of a Regular Pentagon
The interior angle of a regular 36-gon
If the interior angle is 188,° how many sides?
The sum of interior angles of a 22-gon?
The sum of exterior angles of a 173-gon?