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

Math 409/409GHistory 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 Angles3, 4, 5, 6

Alternate Interior Angles3 & 6; 4 & 5

Exterior Angles1, 2, 7, 8

Alternate Exterior Angles1 & 8; 2 & 7

8765

4321

If alternate interior angles are equal when two lines are cut by a transversal then the lines are parallel. (P1.27)

Given: 1 = 2Prove: AB ІІ CD

E

1

2

A B

C D

F

We will prove this by way of contradiction by assuming that the lines are not parallel.

Given: 1 = 2Prove: AB ІІ CD

E

1

2

A B

C D

F

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.

F

E

DC

BA 1

2 2G

A B

C D

E

F

1

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

1

F

E

DC

BA

G2

1 > 2

But this contradicts the hypothesis that 1 = 2.

Given: 1 = 2Prove: AB ІІ CD 1

F

E

DC

BA

G2

1 > 2

E

1

2

A B

C D

F

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 5If 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 180o, then the two lines intersect at a point on the side of the transversal where the interior angles are less than 180o.

1 + 2 < 180

E

F DC

BA

2

11

F

E

DC

BA

G2

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 180o.

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 180o. (2 + 3 180o)

1

F

E

4

DC

BA

2

3

Proof of P1.29a

Given: AB ІІ CDProve: 1 2

By way of contradiction, assume that 1 2. Without loss of generality, assume that 2 < 1.

1

2

A B

C D

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 < 1Prove:

1

2

A B

C D

Given: 2 < 1

Prove:

Statement Reason2 + 3 < 1 + 3 Given, CN 11 + 3 = 180o P1.132 + 3 < 180o CN 1AB and CD intersect Ax. 5

Def ІІ

3

2

1A B

C D

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 CDProve: 4 2

Statement ReasonAB ІІ CD Given1 2 P1.29a4 1 P1.134 2 CN 1

4

DC

BA 1

2

Proof of P1.29c

Given: AB ІІ CDProve: 3 + 2 180o

Statement Reason AB ІІ CD Given 4 2 P1.29b 3 + 4 180o P1.13 3 + 2 180o CN 1

4

DC

BA

2

3

Our last proposition deals with parallelograms.

Definition of a parallelogramA parallelogram is a quadrilateral in which opposite sides are parallel.

AD BCAB DC

A B

CD

Opposite sides of a parallelogram are equal. (P1.34)

Given: ABCD is a parallelogram

Prove: AB = DCAD = BC

A B

CD

Statement ReasonConstruct BD Ax. 1AB ІІ DC, AD ІІ BC Def. parallelogram1 2, 3 4 P1.29a

IdentityABD CDB ASA (P1.26a)

Def.

Given: ABCD is a parallelogram

Prove: AB = BC, AD = BC

AB = BC, AD = BC

BD = DB

1

3

4

2

A B

D C

This ends the lesson on

Book I of Euclid’s Elements Part V: Parallel Lines