Question

Can you help me to prove this proposition: If two distinct lines intersect then they intersect in exactly one point.

Answer 1

a line is straight, and continues in one direction forever
they will meet only once and never again

Answer 2

its my geometry prove

Answer 3

?

Answer 4

my teacher said, we need postulates and define for this prove

Answer 5

Existence and Connection
Postulate 1: The collection of all points in geometry forms a set called space.
Definition: Any non-empty set of points in space is known as a geometric figure.
Postulate 2: Lines and planes are geometric figures.
Definitions:
1. We say that two geometric figures are equal iff they represent the same set of points. (i.e. the sets of points are equal as sets.)
2. If a point A is an element of a line and a plane we say that the point is on the line and in the plane and that the line and plane contain the point. We also say that the line goes through the point.
3. We say that a set of points are collinear iff there exists at least one single line containing all of the points in the set.
4. We say that a set of points are coplanar iff there exists a single plane containing all of the points in the set.
Note: The definitions and postulates above allow us to use basic set theory in our description of points. The next several postulates guarantee the existence of at least some points, lines, and planes in the geometry.
Postulate 3: There exist at least four non-collinear, non-coplanar points. (H.I7, K.I5)
Postulate 4: Every plane contains at least three non-collinear points. (H.I7, K.I5)
Postulate 5: Every line contains at least two points. (H.I7, K.I5)
Postulate 6: Given any two distinct points there exists exactly one line that contains both of them. (H.I1, H.I2,B.II, K.1)
Definition: We say that the two points determine the line iff Postulate 6 is true.
Notation: We represent the unique line containing points A and B as AB.
Postulate 7: Given any three distinct non-collinear points, there exists exactly one plane containing them. (H.I4, H.I5)
Definition: We say that the three non-collinear points determine a plane iff Postulate 7 is true.
Notation: We represent the unique plane containing non-collinear points A, B and C as plane ABC.
Postulate 8: If two points lie in a plane, then the line containing them lies in the plane. (H.I5, K.I3)
Postulate 9: If two planes intersect then their intersection is a line. (~H.I6, K.I4)

Answer 6

You can prove by contradiction of Postulate 6.

Assume the two DISTINCT lines intersect at two points. Then those two points must BOTH lie on the two lines. That means there are two different lines passing through the two points which contradicts Postulate 6. Therefore, two straight lines can intersect at most in just one point.

Answer 7

Thanks, but how can I start the proof by 2 columns table???

Answer 8

my teacher wants a proof like this