Question

Lara wrote the statements shown in the chart.
(Statement)
Description

(1)
If two lines intersect, then they intersect at exactly one point.

(2)
In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the length of the legs.


Which option best classifies Lara’s statements?

Statement 1 and Statement 2 are postulates because they are true facts.

Statement 1 is a theorem and Statement 2 is a postulate.

Statement 1 is a postulate and Statement 2 is a theorem.

Statement 1 and Statement 2 are theorems because t

Answer 1

(2) is a fancy way of saying that thing you've heard over and over again
\[a^2 + b^2 = c^2\]

Do you know what that's called?

Answer 2

actualy bouth statments are not necesarly true, :)

Answer 3

But in this case the most sweetable answer is:
Statement 1 and Statement 2 are postulates because they are true facts
i think

Answer 4

Why do you say that, Myko? Some non-Euclidean geometry?

Answer 5

just take this figures on spherical surface ...

Answer 6

Yeah, well. That's not really relevant. But sure.

Answer 7

so i am not sure about saying " they are true facts"

Answer 8

She's a high school geometry student. Bringing up spherical geometry isn't serving any purpose but confusing the issue. For now, it's fine for her to consider them to be true.

Answer 9

They are both theorems since both of them can be proved. Postulates are statements taken for granted.

Answer 10

you right...

Answer 11

First statement seems axiomatic to me...

Answer 12

The axiom is that "For any two points, there is exactly one line containing them."

Answer 13

Although it does seem like they could be seen as equivalent.

Answer 14

I'd call it equivalent.

Answer 15

To prove that statement, would you rely on any axiom BESIDES that one? And if not, then wouldn't we say it's simple a restated axiom? Meh, I dunno.

Answer 16

You would only rely on that axiom. But does that mean they're both postulates? Or is one of them a postulate and the other a theorem?

Answer 17

If they're equivalent statements, it seems arbitrary to call one of them a postulate and the other a theorem.

Answer 18

What about the axiom of choice and Zorn's Lemma? Both are equivalent, yet one of them is called a lemma.

Answer 19

Unless it's a P implies Q
but Q does not imply P situation...

Answer 20

You would need at least two axioms to prove it. So I will continue to think it is a theorem.

Answer 21

I think it's a biconditional though.
And I'm not sure, but I will say that the use of the word "Lemma" usually has to do with the intended USE of a statement. That is, it's used as an in-between statement that gets me to a more interesting theorem.

Answer 22

Oh? I thought you agreed earlier that it only takes the one axiom...

Answer 23

It seems I've found a few alternate proofs online. The most thorough requiring two axioms. Although the first axiom is arguably a definition.

Answer 24

That irks me. =/ You may be right though. I'm surely unconvinced.

Answer 25

Postulate 1: A line contains at least 2 points.
Postulate 2: Through any two points there is a unique line.

Answer 26

I am also unconvinced, but for now, I think it's a theorem.

Answer 27

At least 2 points? lolwut?

Answer 28

the answer was the last one, Statement 1 and Statement 2 are theorems because t

Answer 29

hahahaha