Fifth Postulate
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Contents |
Statement
If a straight line falling on two straight lines make the interior angles on the same side
less than two right angles, the two straight lines, if produced indefinitely, meet on that side
on which are the angles less than the two right angles.
Proof
Parallel lines
First, draw a line AB as shown in figure such that AB and CD bisects each other.
(1)$\angle 4 = \angle 5$ (Vertically opposite)
(2)OA = OB (Construction)
(3)OD = OC (Construction)
(4)$\triangle OAD \cong \triangle OBC$ (SAS)
(5)$\angle 2 = \angle 3$ (CPCT)
(6)$\angle 1 + \angle 2 = 180^o$ (Linear pair)
(7)$\angle 1 = \angle 3 = 180^o$ (from step (5)
Non-Parallel lines
In this case,draw a line AB $\perp$ to any one line and bisector to both lines.
Let us also suppose that $\angle 2 = \angle 3$
(1)OA = OB
(2)$:\angle 2 =\angle 3 $ (Supposition)
(3)OC = OD(Construction)
(4)$\triangle OCB\cong\triangle ODA$(SAS)
(5)$\angle 4 = \angle 5 $ (CPCT)
(6)But $\angle 5 \ne 90^0$ as AB can be $\perp$ to only one line and this contradicts
the supposition that $\angle 2 = \angle 3$
So $\angle 1 + \angle 2 = 180^0$ (Linear pair)
and $\angle 2 \ne \angle 3$
Therefore $\angle 1 + \angle 3 \ne 180^0 $
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