Proving Congruence of Geometric Figures with Superposition

[chemistry1]

Show that if a geometric figure is congruent to another geometric figure, which is in its turn congruent to a third geomtric figure, then the first geometric figure is congruent to the third.


Answer : I will be showing what the question asks by using superposition of the geometric figures (Wether it's in 1D,2D, or 3D)
First, we have 3 geometric figures : A, B, and C.

And also, we know that A is congruent to B and B is congruent to C.

So, : A==>B==>C

Now, let's show that A is congruent to B, that is : A==>C
We know that A is congruent to B and C is congruent to B. So this gives us two possible answers :

C is congruent to A or C is not congruent to A. Let's see what would happen if C wasn't congruent to A.
We know that two things which are equal to the same thing are equal together. A is congruent to B. C is congruent to B. But we said that C couldn't be equal to A. Again, we said that two things which equal the same thing are equal together. We now have a contradiction, so this is not the possible answer. The last choice is : C is equal to A, because A is congruent to B, C is congruent to B. So, two things which are equal to the same thing are equal together.

*I'm taking for granted that : Two things which are equal to the same thing are equal together. Will you accept it or not ?

Thank you and please tell me what you think.(I'm kinda new to it)

 

[mfb]

We know that two things which are equal to the same thing are equal together.

If you know that, you can directly apply that principle and you are done. No need to make all those additional statements.

*I'm taking for granted that : Two things which are equal to the same thing are equal together. Will you accept it or not ?

No, I would consider the proof of that as equivalent to your homework problem.

 

[chemistry1]

If you know that, you can directly apply that principle and you are done. No need to make all those additional statements.

No, I would consider the proof of that as equivalent to your homework problem.

Ok, well I invented it because I thought it was something obvious^^

 

[verty]

Am I right in thinking that this is for practice writing proofs? You started by saying you would use superposition of figures but I don't see that in your argument. I see A = B = C, therefore A = C. In this case I won't accept this for the figure as a whole.

I want to see something like this: figures are congruent if..., now translating A onto B and C onto B, we see that...

 

[chemistry1]

Ok, I'll write something better. Thank you.

 

[chemistry1]

By superposistion, we will prove that if : A ==> B ==> C then A==>C If A is congruent to B, and B is congruent to C, then A is congruent to C.
We know that A is congruent to B and B is congruent to C. We have three lines A(a-b),B(c-d),and C(e-f).Btw, the letters in the parentheses are the points on the end of each line.

(By superposition, and with the information given, point a will go on c and point b will go on d. This results in having line A the same length as line B
By superposition, and with the information given, point e will go on c and point f will go on d. This results in having line c the same length as line B) I'm not sure if I should have showed it, even if it was given by the question that they were congruent.

Conversely, if we were to superpose B on line A and C, and with the information given and what was proved, it should fit exactly, because if it didn't, it would be absurd that a line A-C which fits in B wouldn't do the same inversely, which means that our lines wouldn't be equal. But we proved that they were, so it would be a contradiction.

So, if line A and C have the same length as line B, then this means that line B will also have the same length as line A and C, which means our 3 lines are equal. If we take line A and C, and put point a on point e, and put point b on point f, we see that the lines make one line and are equal together.

 

[mfb]

What do you mean with superposition? Superposition of what, how?

We have three lines A(a-b),B(c-d),and C(e-f).

You have arbitrary geometric shapes, they don't have to be lines.

point a will go on c and point b will go on d.
[...]
By superposition, and with the information given, point e will go on c and point f will go on d.

Okay, the labels are arbitrary, you can do that.

This results in having line A the same length as line B
This results in having line C the same length as line B

Okay, now combine both to compare A with C.

Conversely, if we were to superpose B on line A and C, and with the information given and what was proved, it should fit exactly, because if it didn't, it would be absurd that a line A-C which fits in B wouldn't do the same inversely, which means that our lines wouldn't be equal. But we proved that they were, so it would be a contradiction.

So, if line A and C have the same length as line B, then this means that line B will also have the same length as line A and C, which means our 3 lines are equal. If we take line A and C, and put point a on point e, and put point b on point f, we see that the lines make one line and are equal together.

Just more complicated ways to express things you already said.

Now extend the proof to arbitrary geometric shapes.

 

[chemistry1]

I mean by superposition when you put one thing on another thing.

 

[chemistry1]

Ok, I tried to make it in the simplest way.

We have 3 geometric figures of the same kind.(ex :3 right triangles, 3 lines, etc.)
Let's name them A,B, and C. The question also says that : A is congruent to B, C is congruent to B.
The figure B must be congruent to the two figures A and C, because if it wouldn't, this would mean that they aren't equal, but we are given that they are(A to B, and C to B), so it would be a contradiction.

Thus, we have 3 equal geometric figures. So, because A,B, and C are equal, we can conclude that A is congruent to C.