There are 2 questions attached.
Name the inequality theorems that supports these statements.
I am not getting the name of these theorems. please help.
For problem 1, are you looking for something like this?
The triangles are congruent if sides TS = WV. In that case the angles TRS and WUV are equal. However TS > WV so we may say that angle TRS is greater than WUV.
It works, but I don't like this argument for the following reasons:
1) It rests on the requirement that angles stay the same as we go from one triangle to another and there is no a priori reason to assume this.
2) The proof of this argument is likely going to rely on using the Law of Cosines, as CaptainBlack suggested.
So I agree with CaptainBlack's comment.
-Dan
I google searched it and found it like this.
INEQUALITIES IN TWO TRIANGLES
THEOREM : SSS INEQUALITY THEOREM: In two triangles with two sides congruent, but the third sides not congruent, then the smaller included angle is opposite the smaller side.
THEOREM : SAS INEQUALITY THEOREM: In two triangles with two sides congruent but included angles not congruent, then the third sides are not congruent, and the smaller side is opposite the smaller included angle.
So here, first is SSS INEQUALITY THEOREM
and
Second is SAS INEQUALITY THEOREM.
You can find about this theorems here.
It would be a good guess that the text material follows terminology invented by Edwin Moise in his Elementary Geometry From an Advanced Standpoint. Most mathematics educators in North America have used that text. In his chapter on geometric inequalities theorem 6 is The Hinge Theorem. If two sides of one triangle are congruent respectively to two sides of a second triangle and the included angle in the first triangle is greater than the included angle in the second triangle then the opposite side of the first is greater than the opposite side of the second.
This has a purely synthetic proof and its converse follows at once.