# Could the congruence of these two triangles be proven?

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.

• I guess that you mean $XB=YW,\angle{XBN}=\angle{YWZ}$. But don't we have any other conditions? – mathlove Oct 16 '14 at 11:08
• 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! – konewka Oct 16 '14 at 11:57

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 $$XN_1=XN_2,\ \ \angle{XBN_1}=\angle{XBN_2},\ \ BN_1\not=BN_2.$$

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.