Download MAT 360 Lecture 6 - Stony Brook Mathematics

MAT 360 Lecture 6
Hilbert Axioms
Congruence
To come
MLC
 Sketchpad projects
 Midterm – 4 problems
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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
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CONGRUENT
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This term will be used in two ways:
1. Segment CD is congruent to segment EF
2. Angle <A is congruent to angle <B
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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
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Prove that
segment AB is congruent to segment BA
 If AB ~ CD then CD ~ AB
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Congruence Axiom 3
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If
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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
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<BAC < B’A’C’
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Congruence Axiom 5
If <A ~ <B and <A ~ <C then <B ~ <C.
 <A~<A
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Proposition
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If <A ~ <B then <B ~ <A
Definition
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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.
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NOTE: This is third use of the word
“congruent.”.
Congruence Axiom 6 (SAS)
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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
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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.
Definition
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The symbols AB<CD mean that there
exists a point E between C and D such
that AB~CE.
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The symbols CD>AB have the same
meaning.
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Proposition
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Exactly one of the following conditions
holds
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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.
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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.
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Definition
Suppose that there exists a ray EG
between ED and EF such that
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<ABC ~ <GEF.
 Then we write <ABC < <DEF.
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Proposition
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Exactly one of the following holds
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<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
 If <P <<Q and <Q<R then <P<<R
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(typo)
Proposition (SSS)
Given triangles ΔABC and ΔDEF. If
AB~DE, BC~EF and AC~DF then
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ΔABC~ ΔDEF
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Note: from now on, in the slides, we
denote congruence by ~
Proposition
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All right angles are congruent with each
other.