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Middle/High School Algebra, Geometry, and Statistics (AGS)
5.10 Theorem: Distinct Lines

Axiom - 1:
Given any two points in a plane, there exists one and only one line containing them.

Theorem - I:
Two distinct lines cannot have more than one point in common.

Hypothesis:
'l' and 'm' are two distinct lines that is l m.

Conclusion:
'l' and 'm' cannot have more than one common point.

Proof:
Let us assume that 'l' and 'm' have two common points P and Q contrary to conclusion.
Then P and Q determine the line 'l' and also the line 'm'.
But by Axiom - 1, there exists one and only one line containing two given points.
So 'l' and 'm' cannot be distinct.
This contradicts the Hypothesis.
So, our assumption that 'l' and 'm' have more than one point in common is false.
Hence, we conclude that 'l' and 'm' cannot have more than one point in common.


Directions: Draw two distinct lines and prove that they cannot have more than one point in common.


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