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Consider the Cartesian plane with the standard notions of lines, incidence and betweenness, congruence of angles, and the absolute value distance function.

A) Does the side-angle-side model for congruence hold? If not show a counter example.

B) Give an example of an equilateral triangle with 3 distinct angles.

For part A) I know it does not hold but I cannot prove why. Part A) is similar to exercise 9.3 in Geometry: Euclid and Beyond on page 96. Link is below:

https://books.google.ca/books?id=C5fSBwAAQBAJ&printsec=frontcover&dq=euclid+and+beyond&hl=en&sa=X&redir_esc=y#v=onepage&q=euclid%20and%20beyond&f=false

Thanks for the help.

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    What is the standard notion of lines etc. in the Cartesian plane? – 2017-02-27
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    They are referring to the axioms of incidence (I1) to (I3) and axioms of betweeness (B1) to (B4). Which are written in the text – 2017-02-27
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    Look, if it is a well defined set of points then you have to define the other concepts as well. You cannot just say that "these are undefined things." – 2017-02-27
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    BTW: page 96 is not available for free. You, please give more context. – 2017-02-27

1 Answers 1

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Take triangles $AOB$ and $COD$, where $O=(0,0)$, $A=(1,4)$, $B=(4,-1)$, $C=(2,3)$ and $D=(3,-2)$. Using taxicab geometry to compute distances (that's what you mean by "the absolute value distance function", as I discovered in Hartshorne's book) you have: $OA=OB=OC=OD=5$ and of course $\angle AOB=\angle COD=90°$.

But those triangles are not congruent, because $AB=8$ and $CD=6$.

For part B) you can try for instance the triangle with vertices $(0,0)$, $(1,5)$ and $(4,2)$.

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    @j.stat Have a look at (http://jwilson.coe.uga.edu/MATH7200/TaxiCab/TaxiCab.html) – 2017-02-27