Prove Euclid's Fifth Postulate using Triangle Sum?

Expert Solution

Euclides fifth postulates States that " if a line segment intersect two lines forming two interior angle on the same side that sum to less than two right angle , then the two lines , if extended indefinitely , meet on that side on which the angles sum to less than two right angle".

We can prove it by contradiction method.

Let AB and CD be two straight lines and another straight line L meet with AB at E and With CD at F .

Where <BEF+<DFE is less than 180°.

We have to prove that if AB and CD are extended indefinitely then AB and CD wil meet on the side in which <BEF and <DFE occurs.

Assume that the two lines AB and CD will meet on the side in which <AEF and <CFE occurs , and the meeting point is M.

Now <BEF+<DFE is less than 180° implies that <AEF+<CFE is greater than 180°.

Then from triangle EMF we have <EMF=180°-(<MEF+<MFE) =180°- (angle greater than 180°) = negative value. But any Angle can not be negative. So,our assumption was wrong . So by contradiction method two lines meet on the side in which <BEF and <DFE occurs. And if AB and CD meet at N then<ENF=180°-(<NEF+<NFE).

This completes the proff.

w

milcah answered 2 years ago

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