geometry

[John Doe]

I have completely forsaken you guys (you weren't posting for so long), so I feel guilty about spamming you with homework questions. But feel free (math brains) to take this as a challenge!

How would you prove that:

If the diagonals of a trapezoid are congruent, then it is an isosceles trapezoid.

I need to do a two-column, but you don't need to do it all. Just a plan for it. Thanks in advance.

[Tarek]

DA FUQ!? I have a 102 in geometry. Dont meen i like doing it. And i think i speak for most people on this group when i say, no one wants to do an unneeded proof. Sry.

[John Doe]

ok...

I thought people might enjoy the CHALLENGE!

[John Doe]

Please...

Whether to show off skill, help out a guy, or improve, doesn't anyone want to give it a whirl?

[Silence Dogood]

I think the last time I got that problem I grew lazy and just put a question mark on my hw. Sorry! However, I'll ponder it once again

[Silence Dogood]

Well it has to be something along the lines of the inside angles formed by the angles are congruent by vertical angles and then you do something with the diagonals being congruent to prove those two triangles congruent, thus proving that the legs are congruent by CPCTC, thus proving it is isoceles

[John Doe]

Oh thank you very much. 

[John Doe]

CPCTC

what is that?

coresponding parts of congruent triangles c------?

[Richard Copenhaver]

congruent

c oresponding parts of congruent triangles congruent

[Silence Dogood]

Corresponding parts of congruent triangles are congruent. Triangle ABC is congruent to triangle DEF so angle A is congruent to angle D, andle E is congruent to angle E, etc.

[John Doe]

ohh ok

[John Doe]

I see but the problem is proving the triangles congruent

tell me which theorem to use:

http://upload.wikimedia.org/wikipedia/commons/5/5e/Isoscelestriangle2.svg 

SSS

SAS 
AAS

ASA

AAA (just looking to see if you were paying attention)

[John Doe]

Awww... Hello?

[Richard Copenhaver]

side angle side maybe hmmm

[John Doe]

I see which triangles? (use the wiki article if necessary)

[Richard Copenhaver]

BEA CED
because its given that the diagonals are congruent

where they bisect each other would be congruent 

those segments i mean 

so you have your two sides and angle E would be your congruent angle but i'm not quite sure how you would prove E to be congruent to E

[Richard Copenhaver]

oh wait

are a linear pair i think

thats what it's called right

*geometry fail

no but because the segments in the trapezoid are parallel

they would be congruent because they are a linear pair formed by the transversal 

[John Doe]

Thanks a lot. I have been exploring similarity as well. I know the theorem that if the base angles are congruent then it's isoc for trapezoids.

[Richard Copenhaver]

and the transversals are the diagonals

[John Doe]

Ok gotcha. Thanx. Our geometry teacher says if we don't prove a theorem we can't use it on ANY test or hw (we get a zero for that problem). ikr?

[Richard Copenhaver]

yes also if you proved the triangles congruent then you could use CPCTC to say that the legs are congruent to each other

through the therom about iso. trapezoid legs

[John Doe]

Here's the problem though, the diagonals don't bisect eahcother

[Richard Copenhaver]

intersect

i mean

not bisect

they intersect a E

[John Doe]

I see but what theorem would you use to show they split into congruent segments ie. be=ce etc.?

[Silence Dogood]

The one that John suggested was the one I was thinking of. SAS, that's why I said vert. ang. thrm.