Geometry
I did not get this question at all, if someone could help, I would appreciate it!
It says:
Reword Theorem 1-3 as two statements, one describing existence and the other describing uniqueness.
Theorem 1-3 says:
If two lines intersect, then exactly one plane contains the lines.
Earlier in the lesson it explained something about existence (there is at least one point of intersection) and uniqueness (no more than one such point exists) about Theorem 1-1, but I still couldn't put the two together. Can anyone help??? I need this homework turned in tomorrow, or it's late. Any help is appreciated! THANKS!!!
you may have already proven that 3 points determine a plane.
If two lines intersect, then pick the point of intersection and two other points, one on each line. Those 3 points determine a plane.
google can provide you with other more rigorous proofs.
Sure! Theorem 1-3 states that if two lines intersect, then exactly one plane contains the lines. To reword this theorem as two statements, we can separate it into one statement describing existence and another describing uniqueness.
Existence: If two lines intersect, then there exists at least one plane that contains the lines.
This statement affirms that there is at least one plane that contains the intersecting lines.
Uniqueness: If two lines intersect, then no more than one plane contains the lines.
This statement states that there cannot be more than one plane that contains the intersecting lines.
By rephrasing Theorem 1-3 in this way, we have described both the existence and uniqueness aspects related to the intersecting lines and the containing plane.
To reword Theorem 1-3 as two statements, one describing existence and the other describing uniqueness, we need to break down the original theorem and focus on its different aspects.
The original theorem, Theorem 1-3, states: "If two lines intersect, then exactly one plane contains the lines."
To reword it, let's first consider the existence aspect. Existence refers to the fact that there is at least one point of intersection between the lines. In Theorem 1-1, it was stated that two distinct lines in a plane always intersect at exactly one point. So, we can reword the existence part of Theorem 1-3 as: "If the lines intersect, there exists at least one point of intersection."
Now, let's move on to the uniqueness aspect. Uniqueness refers to the fact that there is no more than one point of intersection. In Theorem 1-1, it was also mentioned that two distinct lines can't intersect at more than one point. So, we can reword the uniqueness part of Theorem 1-3 as: "If the lines intersect, there is at most one point of intersection."
To summarize, Theorem 1-3 can be reworded as two statements:
1) Existence: "If the lines intersect, there exists at least one point of intersection."
2) Uniqueness: "If the lines intersect, there is at most one point of intersection."
Remember to understand the original theorem and its concepts in order to reword it accurately.