How Many Congruences Exist for Isosceles Triangle ABC?

[mathstudent88]

Given a triangle ABC which is isosceles but not equilateral. That is, AB = AC, but AB does not equal BC. How many congruences are there, between triangle ABC and itself?

Here's my answer:

By the hypothesis, we can infer that triangle ABC is congruent to triangle ACB. So there is just one congruence.

My professor said that there is another congruence but I just can't figure out what it is. Can someone please help me?

Would it be that triangle ABC is congruent to triangle ACB which is congruent to triangle BAC?

Thank you!

 

[Dick]

I don't know what your professor is talking about. ABC is congruent to ACB. That's about it. BAC is not congruent to ABC since AC is not equal to BC.

 

[mathstudent88]

I do not know what my professor is talking about either. She gave me back my homework and told me to re-do this one and she wrote on my paper that there is another congruence but I am just not seeing it. She put on my paper, "Actually, there is one other."

Do you have any ideas?

Thanks for the help!

 

[Dick]

Could she mean ABC is congruent to ABC? It's kind of obvious, but it is true (for ANY triangle).

 

[mathstudent88]

I sent her an e-mail asking her about it and this is what she said to me:

"Here's a hint: It is the trivial one, i.e. the one that maps each vertex to itself."

Do you have any idea what she means by that? ABC congruent to ABC?

 

[Dick]

Sure. It's just the obvious statement that any triangle is congruent to itself. AB=AB, BC=BC, CA=CA.

 

[mathstudent88]

Haha ok. Thanks soo much for the help! :)