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Using any of the known theorems(SSS, ASA, SAS, HL), Could it be proven that these two triangles(XBN, YWZ) are congruent?

*Given: XB=YW,∠XBN=∠YWZ and XYZN is a rectangle. enter image description here

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  • $\begingroup$ I guess that you mean $XB=YW,\angle{XBN}=\angle{YWZ}$. But don't we have any other conditions? $\endgroup$ – mathlove Oct 16 '14 at 11:08
  • $\begingroup$ The result of your other post can't be used here, since $BN$ and $ZW$ don't lie on the same line, as the commenters on your other question also pointed out! $\endgroup$ – konewka Oct 16 '14 at 11:57
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We only have $$XB=YW,\ \ \angle{XBN}=\angle{YWZ},\ \ XN=YZ.$$ So, by this, we cannot say these two triangles are congruent.

Counterexamples :

(1) Draw the line $BX$.

(2) Draw a circle, whose center is $X$, whose radius is smaller than $BX$.

(3) Draw a line, which passes through $B$, which crosses the circle $X$ at two points. Call them $N_1,N_2$.

Then, we have two triangles $\triangle BXN_1,\triangle BXN_2$ such that $$XN_1=XN_2,\ \ \angle{XBN_1}=\angle{XBN_2},\ \ BN_1\not=BN_2.$$

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As addition to mathlove's answer, here is an explicit counterexample, which has the exact properties you state, but it should be clear that the triangles are not congruent. enter image description here

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