MAT 360 Lecture 6

Hilbert AxiomsCongruence

To come MLC Sketchpad projects Midterm – 4 problems

Models and interpretation. Proof from Hilbert’s axioms Produce a definition of some known object Definitions of terms we learn (like

independence, categorical) will not be asked directly but “applied”

Congruence Axiom 1 If A and B are distinct points then for any

point A’ and for each ray r emanating from A’ there exist a unique point B’ on r such that B’≠ A’ and AB ~ A’B’.

Recall we have an undefined term CONGRUENT

This term will be used in two ways:1. Segment CD is congruent to segment EF2. Angle <A is congruent to angle <B

Question: Could we use different words for the use 1. and the use 2?

Congruence Axiom 2 If AB ~ CD and AB ~ EF then CD ~ EF AB ~ AB

Prove that segment AB is congruent to segment BA If AB ~ CD then CD ~ AB

Congruence Axiom 3 If

A*B*C, A’*B’*C’, AB ~ A’B’ BC ~ B’C’

Then AC ~ A’C’

Congruence Axiom 4 Given an angle <BAC, a ray A’B’ and a

side of the line A’B’ there is a unique ray A’C’ emanating from the point A’ such that

<BAC < B’A’C’

Congruence Axiom 5 If <A ~ <B and <A ~ <C then <B ~ <C. <A~<A

Proposition If <A ~ <B then <B ~ <A

Definition Two triangles are congruent if there is a

one to one correspondence between the vertices so that the corresponding sides are congruent and the corresponding angles are congruent.

NOTE: This is third use of the word “congruent.”.

Congruence Axiom 6 (SAS) If two sides and the included angle of a

triangle are congruent respectively to two sides and the included angle of another triangle then the two triangles are congruent.

Proposition Given a triangle ΔABC and a segment DE

such that DE~AB there is a unique point F on a given side of the line DE such that the ΔABC~ΔDEF

Proposition If in ΔABC we have that AB~AC then

<B~<C.

DefinitionThe symbols AB<CD mean that there exists

a point E between C and D such that AB~CE.

The symbols CD>AB have the same meaning.

Proposition Exactly one of the following conditions

holds AC<CD, AB~C or AB>CD

If AB<CD and CD~EF then AB<EF. If AB>CD and CD~EF then AB>EF. If AB<CD and CD<EF then AB<EF.

More Propositions Supplements of congruent angles are

congruent. Vertical angles are congruent to each

other An angle congruent to a right angle is a

right angle. For every line l and every point P there

exists a line through P perpendicular to l.

Definition Suppose that there exists a ray EG

between ED and EF such that <ABC ~ <GEF.

Then we write <ABC < <DEF.

Proposition Exactly one of the following holds

<P < <Q , <Q < <P or P ~ Q. If <P<<Q and <Q~<R then<P <<R If <P ><Q and <Q~<R then<P > <R (typo)

If <P <<Q and <Q<R then <P<<R

Proposition (SSS) Given triangles ΔABC and ΔDEF. If

AB~DE, BC~EF and AC~DF then ΔABC~ ΔDEF

Note: from now on, in the slides, we denote congruence by ~

Proposition

All right angles are congruent with each other.