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Prove the Converse of Proposition 3.3 by using Betweenness Axiom 1. The converse is Given B...

Prove the Converse of Proposition 3.3 by using Betweenness Axiom 1. The converse is Given B ? C ? D and A ? B ? D, then A ? B ? C and A ? C ? D. Please do not use "by mapping of letters"

Expert Solution

(figure)

Given: and

To prove: and

Proof: (Method 1)

Now Betweenness Axiom 1: If , then A,B, and C are three distinct points all lying on the same line, and

According to the converse of proposition 3.3 If , then B, C and D are three distinct points.

We can infer from this that Pt C lies between B and D .....................................(1)

If , then A, B and D are three distinct points and Pt B lies between Pts A and D.

Now, as Pt C lies between B and D and Pt B lies between A and D. So it means that points B and C lies between A and D.

From statement 1, we could infer that it is Pt B which lie between A and C.

......................(2)

Now can also be ...............(3)

From 2 and 3, we can infer

Thus implies Pts A, B, C and D are four distinct points.

Method 2:

If then .......................(1) (Betweenness property)

and then ...................(2)

From1 and 2 ,

It means that Pts A, B , C and D are four distinct points such that

Thus proved

milcah answered 1 year ago

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