Don't know if this will help, but this is how i would do ithope it helps
I have tried and tried and can't figure out this one. I actually cannot figure out where to start. I need to prove that triangle AGC is an isosceles triangle.
I need to prove it in 10 steps and so far, I have:
angle BGE is congruent to angle DGE becuase it was given.
I believe that the last step in my proof will be that triangle AGC is isosceles because the definition of isoscels is that 2 angles or 2 sides are eaqual.
I just can't figure out fow to connect the begining of the proof to the end. (By proving that which 2 sides or 2 angles are congruent.)
Any help will be greatly appreciated!
Thanks in advance!
in case it is needed, I was also given that:
line GB is congruent to line GB,
angle BGE is congruent to angle DGE
line GF is perpendicular to line AC
a picture of my triangle is attached
Thank you Jacs for your speedy reponse.
If I understand your response correctly, the only way to prove it is by assuming that the angles at the top of the triangle (angles AGF and CGF) are equal. We assume this becuase they have the arch ark through them?
I was thinking this too, but then I wasn't sure that this was an acceptable line of reasoning, since it wasn't sepcifically "given" that angle AGF and angle CGF are equal.
Is it a generally accepted policy to assume these angles are equal since there are identical arch marks through them?
You are welcome
Yes those arcs, if they are identical, mean that the angles are equal, and so techincally it is not an assumption, it is a piece of info provided to help you work it out.
If one had an arc, and one was a double arc or something diffferent, that would indicate that they were not equal.
As for the only way to prove it, that is not the case at all, there are probably at least 5 comparable ways which would all be just as correct.
geometry often has many routes to get to the same destination, often they are all correct, some just longer and more complex than others. My rule of thumb is easier is better, so i always go for the quickest method I can see.
Of course, you don't always see the short way.
So this would just be one model of how to approach the question, don't take it as gospel though, i am sure there are many other alternatives that woudl yeild the same result.
Thank you Jacs, for the clear explanation.
If you wouldn't mind explaining another way to prove that the triangle is isosceles without using the fact that angle AGF and angle CGF are congruent, it would be appreciateld greatly.
Woah! please ignore this last response. As I was reviewing your response, I saw my mistake.
I am not sure i can see a way of doing it without using that fact. The only other ways i can see all involve it. I will include an alternative, but i still have to use that fact of the equal angles.
this proof is much more complex though
lol i see you have told me to ignore your last response, but i figure sicne i have already done the solution, i will put it up anyway, so you can ignore it if you choose (it is a yucky more complex one anyway)
Thanks Jacs, for bearing with me. Math is really hard for me and when I think I'm kind of getting it, I like to keep going...so I have another question. I am so excited, I am actually "getting" this a little bit! Yeah! Thanks!
I will prove that angle AGF = angle CGF because they were given
I will also prove angle GAF is equal to angle GCF
With 2 angles being proven to be congruent, I now have enough information to prove the traingle is an iscsceles.
My question is, how do I prove that angle GAF is equal to angle GCF?
I know that the sum of angles in a triangle is 180 degrees.
I also know that angle GFC is 90 degrees and so is it's opposite angle-GFA.
I know that, some kind of way, this means that GAF is congruent to GCF. But, why? Which theorem/postulate/reason? How do you bridge this fact?
I looked at your reasoning and I really appreciate it. When I first began this problem I looked t doing it along those same lines, but I couldnt figure out how to reason it out that way. It really helps me to see that there is way to prove it that way.
can't u just prove triangle AGC congruent triangle CGF by ASA ((what is given, angle AGF congruent angle CGF (given), GF congruent to GF (reflexive), and angle GFA and angle GFC congruent to each other because perpendicular line makes congruent adjacent angles (90 degrees) , so therefor the triangles are congruent, and ag is congruent to CG, making triangle ACG isos. that really isn't 10 steps... more like 4 if you stuff some in x.x
Oh , and jacs, your so wrong
omg, the ANSWER was basically GIVEN to you x.x.... that proof had all the needed info in the given x.x