Question

Show that the AAS and ASA congruence theorems (Euclid I-26) fail in taxicab geometry.

Answer 1

Take A=(0,0), B=(1,1), C=(2,0) then form triangle

Now take D=(0,0),E=(0,1),F=(1,1) form triangle.

let suppose angle BAC= angle EDF=45 degree ; angle BCA=angle DFE=45 degree

also length of AC in taxicab geometry is (2-0)+(0-0)=2;

length of DF IS (1-0)+(1-0)=2;

so sides AC and DF have measure 2 and this gives condition of ASA test but these two triangles are not congruent because

measure of AB IS (1-0)+(1-0)=2

while measure of DE is (0-0)+(1-0)=1 and hence those two are not congruent triangle in Taxicab geometry.

Similarly you can construct triangle for AAS test fallcy in taxicab geometry.