Math 409/409G History of Mathematics Book I of Euclid’s Elements Part V: Parallel Lines

In order to prove anything about parallel lines, we will need the following definitions. Euclid’s Definition of Parallel Lines Two lines are parallel if when produced (extended) infinitely in both directions, they do not meet one another in either direction. Alternate def : Parallel lines never intersect.

Interior and Exterior Angles Interior Angles  3,  4,  5,  6 Alternate Interior Angles  3 &  6;  4 &  5 Exterior Angles  1,  2,  7,  8 Alternate Exterior Angles  1 &  8;  2 &  7

If alternate interior angles are equal when two lines are cut by a transversal then the lines are parallel. (P1.27) Given:  1 =  2 Prove: AB ІІ CD

We will prove this by way of contradiction by assuming that the lines are not parallel. Given:  1 =  2 Prove: AB ІІ CD

Then by the definition of parallel lines, AB and CD intersect. Without loss of generality, assume that the lines intersect at point G on the side of  2.

Then by P1.16, exterior  1 of  EFG is greater than interior  2 of that triangle.

But this contradicts the hypothesis that  1 =  2. Given:  1 =  2 Prove: AB ІІ CD

So our assumption that the lines are not parallel is wrong. (We will prove this by way of contradiction by assuming that the lines are not parallel.) So the two lines must be parallel. This proves that two lines are parallel when alternate interior angles are equal.

We will soon prove Proposition 1.29, the first of Euclid’s propositions needing his fifth axiom. This axiom states: Axiom 5 If two lines are cut by a transversal in such a way that the sum of the interior angles on the same side of the transversal is less than 180 o, then the two lines intersect at a point on the side of the transversal where the interior angles are less than 180 o.

Proposition 1.29 If two parallel lines are cut by a transversal, a)Alternate interior angles are equal. b)An exterior angle is equal to the opposite interior angle on the same side of the transversal. b)The sum of the interior angles on the same side of the transversal is 180 o.

a)Alternate interior angles are equal. (  1   2) b)An exterior angle is equal to the opposite interior angle on the same side of the transversal. (  4   2) b)The sum of the interior angles on the same side of the transversal is 180 o. (  2 +  3  180 o )

Proof of P1.29a Given: AB ІІ CD Prove:  1   2 By way of contradiction, assume that  1   2. Without loss of generality, assume that  2 <  1.

Side Note So our new given (assumption) is now  2 <  1, and we must use thus to show that this new given results in a contradiction of the original hypothesis (Given) that AB ІІ CD. Given:  2 <  1 Prove:

Given:  2 <  1 Prove: Statement Reason  2 +  3 <  1 +  3 Given, CN 1  1 +  3 = 180 o P1.13  2 +  3 < 180 o CN 1 AB and CD intersect Ax. 5 Def ІІ

So our assumption that the alternate interior angles are not equal is wrong. (By way of contradiction, assume that  1   2.) So we must have that  1   2. This proves that alternate interior angles are equal when two lines are parallel are cut by a transversal.

Proof of 1.29b Given: AB ІІ CD Prove:  4   2 Statement Reason AB ІІ CD Given  1   2 P1.29a  4   1 P1.13  4   2 CN 1

Proof of P1.29c Given: AB ІІ CD Prove:  3 +  2  180 o Statement Reason AB ІІ CD Given  4   2 P1.29b  3 +  4  180 o P1.13  3 +  2  180 o CN 1

Our last proposition deals with parallelograms. Definition of a parallelogram A parallelogram is a quadrilateral in which opposite sides are parallel.

Opposite sides of a parallelogram are equal. (P1.34) Given: ABCD is a parallelogram Prove:

Statement Reason Construct BDAx. 1 AB ІІ DC, AD ІІ BCDef. parallelogram  1   2,  3   4P1.29a Identity  ABD   CDBASA (P1.26a) Def.  Given: ABCD is a parallelogram Prove:

This ends the lesson on Book I of Euclid’s Elements Part V: Parallel Lines