Hon. Geometry Fr Chris Thiel, OFMCap 2016

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Presentation transcript:

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

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

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

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

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

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

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

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

Example: #20 p 88

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

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

Remote Interior Angles Theorem

Regular Convex Polygons Not Regular Not Convex

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, …

Sum of Interior Angles of Convex Polygons:

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

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?

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