Question 1210469
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Which  of the following conditions ensures that triangles ABC and DEF are similar? Select all that apply.

(1) A = B, D = C, F = E, B = A, AB/AD = BC/BE
(2) A = D, AB/EF = BC/DE
(3) AB perp CD, DE perp AE, AB = DE and BC = BD
(4) AB parallel to BC, AB parallel to AC, CA parallel to FD
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        The analysis by  @CPhill in part  (1)  is incorrect and his conclusion 

        for this part is incorrect,  too.



<pre>
Indeed,  A = B means that triangle ABC is isosceles with congruent angles A and B at its base.

Next,    F = E means that triangle DEF is isosceles with congruent angles F and E at its base.

The condition D = C means that the angles D and C in these two triangles, ABC and DEF,
opposite to their bases AB and FE, are congruent.


So, the triangles ABC and DEF are similar due to the AAA-test.


That is true that the vertices of these triangles are listed in non-canonic order.
The canonic order should be consistent, like ABC and EFD,
but this does not interfere for triangles ABC and DEF  (or ABC and EFD) to be similar.


The last imposed condition in (1), AB/AD = BC/BE is not relevant to their similarity,
but does not contradict to it.


The similarity of triangles ABC and FED is provided by the conditions on congruency their corresponding angles.
</pre>

So again, &nbsp;in part &nbsp;(1), &nbsp;the analysis and the conclusion by @CPhill are incorrect.