Parallel lines

Parallel lines in Euclidean geometry

Four logically equivalent versions of “the” parallel postulate • That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles. (Joyce’s version of Euclid, Postulate 5)

• If two lines in the same place are cut by a transversal so that the sum of measures of a pair of interior angles on the same side is less than 180, the lines will meet on the side of the transversal. (Kay’s version of Euclid, p 213)

• If l is any line and P any point not on line l, there exists in the plane of l and P exactly one and only one line m that passses through P and is parallel to l. (Playfair’s Postulate, Axiom P-1: Euclidean Parallel Postulate, Kay, p 214)

• If two parallel lines are cut by a transversal, then each pair of alternate interior angles is con- gruent. (Theorem 2, Kay, p 214.)

Using Playfair’s Postulate as the starting point gives a convenient way to characterize the geometries:

If l is any line and P any point not on line l, there exists in the plane of l and P :

• no line though P parallel to l. SPHERICAL

• exactly one and only one line m that passses through P and is parallel to l. EUCLIDEAN

• more than one line through P parallel to l. HYPERBOLIC Transversal Theorems In Euclidean geometry, you can combine the parallel postulate (if lines parallel, then angle relation- ships hold) with its converse “Parallelism in Absolute Geometry” (if angle relationships hold, then lines parallel) to get a collection of “if and only if” statements: l is parallel to m if and only if • same side interior angles are supplementary (the C property).

• corresponding angles are congruent (the F property). • alternate interior angles are congruent (the Z property).

Other major theorems Compare:

An exterior angle of a triangle has angle measure greater than or equal to that of either opposite interior angles. (Absolute - Exterior Angle Inequality)

The measure of an exterior angle of a triangle equals the sum of the measures of the two opposite interior angles. (Euclidean - Exterior Angle Theorem)

The sum of the measures of the angles of a triangle cannot exceed 180. (Absolute - Saccheri-Legendre)

The sum of the measures of the angles of a triangle is exactly equal to 180. (Euclidean - Angle Sum Theorem) Prove the Exterior Angle Theorem and the Euclidean Angle Sum Theorem (Work through the proof in the lecture.) Midpoint Connector Theorem The segment joining the midpoints of two sides of a triangle is parallel to the third side, and has length one half that of the thrid side.

1 LM||BC LM = BC 2

Example: Given AL =5,BL =5,AM =6,BM =6,AC = 18, A − L − B, A − M − C,whatis LM?

Solution: Prove the Midpoint Connector Theorem (Work through the proof in the lecture.)