This discussion makes my eyes bleed and sort of makes me realiZe that people really don't know much math... like at all.
Let's call the sides marked with one line as A, and those marked with two lines as B. Now, ASSUMING the lines in the diagram are straight, the two angles at the center of the figure have to be identical, as they are cuased by the interseccion of two straight lines. Lets say these angles have a value of ALPHA. Now, the Sin Rule says that A/sin(angle opposite A) = B/(sin opposite B) in a triangle. Therefore, since we know the anlge opposite both A's are the same (ALPHA), we therefore conclude that the angles opposibe B are also the same, let's call them BETA.
So then we have two triangles with angles ALPHA and BETA, therefore, since the sum of angles in a triangle is always 180°, the remaining angles must also be the same, let's call them GAMMA.
Now we apply the Cosine Rule: C^2 = A^2 + B^2 - 2*A*B*cos(GAMMA), since A, B, and GAMMA are the same for both triangles, we than have that both have the same other side lenght, C.
In conclusion, we have two triangles with angles ALPHA, BETA, GAMMA and with sides A, B, C, so both triangles are congruent, have a nice day.
