Converse of the Hinge Theorem

[Guest]

Ok.... My text book says:
1)Triangle ABC with median BD; BC>BA. 1)Given
2)AD is congruent to CD 2) Definintion of median of a triangle
3)BD is congruent to BD 3) Reflexive property
4)Therefore the measure of angle BDC is > the measure of the angle BDA
4) Converse of the Hinge Theorem

Now what I don't understand is how they came up with that answer. I would have put the Exterior Angle Inequality Theorem but this is the "correct" answer. Can anyone help me?[/img][/url]

 

[tkhunny]

You do not know anything about the "non-supplementary" angles of either BDA or BDC. This discourages me from liking your answer. Sorry.

The "Converse of the Hinge Theorem" is good, except for one thing. We all should know that the "Converse" is NOT logically equivalent to the original. This gives us a bit of a dilemma. The final step in the "Correct" proof would be valid ONLY IF you have proven the "Converse of the Hinge Theorem". It must have been done in the book, in class, or as part of some assignment. Without this exercise, it is an invalid step. One cannot use an unproven theorem to construct a proof.

Sorry, "Intuitively Obvious" rarely constitutes a valid proof.

 

[pka]

Carefully read the statement of the hinge theorem!
You have $\displaystyle CD \approx DA$ and $\displaystyle BD \approx DB$ so if it were true that $\displaystyle m\angle CDB \le m\angle BDA$ then by the hinge theorem $\displaystyle BA \ge BC$.
But that is contrary to the given

PS
Just out of curiosity, what textbook are you using?
I have seen the name hinge theorem only in work be Edwin Moise.
Moise was an R L Moore student, I am pretty sure he got the name from Moore.
The hinge theorem cannot be found at the MathWorld website