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The Concept of Congruence in Geometry Explained

Learn about congruence axioms, triangle congruence, angle congruence, and their applications in geometry through logical proofs and definitions in this lecture.

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The Concept of Congruence in Geometry Explained

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  1. MAT 360 Lecture 6 • Hilbert Axioms • Congruence

  2. 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”

  3. 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’.

  4. Recall we have an undefined term • CONGRUENT • This term will be used in two ways: • Segment CD is congruent to segment EF • Angle <A is congruent to angle <B • Question: Could we use different words for the use 1. and the use 2?

  5. Congruence Axiom 2 • If AB ~ CD and AB ~ EF then CD ~ EF • AB ~ AB

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

  7. Congruence Axiom 3 • If • A*B*C, • A’*B’*C’, • AB ~ A’B’ • BC ~ B’C’ • Then AC ~ A’C’

  8. 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’

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

  10. Proposition • If <A ~ <B then <B ~ <A

  11. 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.”.

  12. 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.

  13. 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

  14. Proposition • If in ΔABC we have that AB~AC then <B~<C.

  15. Definition • The 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.

  16. 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.

  17. 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.

  18. Definition • Suppose that there exists a ray EG between ED and EF such that • <ABC ~ <GEF. • Then we write <ABC < <DEF.

  19. 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

  20. 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 ~

  21. Proposition • All right angles are congruent with each other.

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