Question

Look at the picture below

John states that by ASA.
Mary states that by AAS.
Phillip states that the triangles are not congruent.

Which student is correct? why?

MY ANSWER:
Phillip is correct. XY=/ AC. (But i have another guess/answer)
No student is right, These triangles arent congruent but these triangles are to different triangles. We can see this with XY/AC. (But doesnt that just mean phillip is correct?) HELP!

Answer 1

Answer: Phillip is correct. The triangles are not congruent.

How do we know this? Because triangle ABC has the 15 inch side between the two angles 50 and 60 degrees. The other triangle must have the same set up (just with different letters XYZ). This isn't the case. The 15 inch side for triangle XYZ is between the 50 and 70 degree angle.

This mismatch means we cannot use the "S" in the ASA or AAS simply because we don't have a proper corresponding pair of sides. If we knew AB, BC, XZ or YZ, then we might be able to use ASA or AAS.

At this point, there isn't enough information. So that means John and Mary are incorrect, leaving Phillip to be correct by default.

Note: Phillip may be wrong and the triangles could be congruent, but again, we don't have enough info. If there was an answer choice simply saying "there isn't enough info to say either if the triangles are congruent or not", then this would be the best answer. Unfortunately, it looks like this answer is missing. So what I bolded above is the next best thing.

Answer 2

Phillip states that the triangles are not congruent.

Explanation:

Assume for a second the triangles are congruent.

Then the angles are equal

<A =< X

<B= <Y

<C = <Z

Then

Triangle ABC = Triangle XYZ

from this information

AB = XY

and BC = YZ

and CA = ZX

but the only other piece of information we have is CA = 15 and XY = 15

If this is true then CA = ZX = 15 and AB = XY = 15

That would make this an isosceles triangle (AB = AC)

so angles B and C would have to be equal, but they are not.

So these two triangles cannot be congruent.