Given A*B*C and A*C*D c) Prove the corollary to Axiom B-4. Axiom B-4: For every

Question

Given A*B*C and A*C*D
c) Prove the corollary to Axiom B-4.

Axiom B-4: For every line l and for any three points A, B, and C not lying on l:
i) If A and B are on the same side of l and if B and C are on opposire sides of l, then A and C are on the same side of l.
ii) If A and B are on opposire sides of l and if b and C are on opposite sides of l, then A and C are on the same side of l.

Corollary (This is what you need to prove): If A and B are on opposire sides of l and if B and C are on the same side of l, then A and C are on opposite sides of l.


Note: Parts a and b to this question were posted as a separate question in order to give more karma points. Please check out that question if you can help and if it'll help you answer this question. Please *thoroughly explain* all steps in the proof to help me understand. Thank you!

Here are the betweenness axioms:
B-1: If A*B*C, then A, B, and C are three distinct points all lying on the same line, and C*B*A.
B-2: Given any 2 distinct points B and D, there exist points A, C, and E lying on line BD such that A*B*D, B*C*D, and B*D*E
B-3: If A, B, and C are three distinct points lying on the same line, then one and only one of the points is between the other 2.

Here are incidence axioms:
I-1: For every point P and for every point Q not equal to P, there exists a unique line l incident with P and Q.
I-2: For every line l, there exist at least 2 distinct points indicent with l.
I-3: There exist three distinct points with the property that no line in incident with all three of them.

Explanation / Answer

If A and B are on opposire sides of l and if B and C are on the same side of l, then A and C are on opposite sides of l.

Proof. We argue by contradiction . By the hypotheses and by the denitions of same and opposite side, neither A nor C can lie on l. Hence, if the two are not on on opposite sides of l, then they must be on the same side. In that case, by Betweenness Axiom , A and B are on the sameside of l. This contradicts the hypothesis that A and B are on opposite sides of l.

Hence A and C are on opposite sides of l.

I hope this helps!

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