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Prove Lemma 1.1.6: For any two distinct lines ℓ1 and ℓ2 in a plane, either ℓ1||ℓ2...

Prove Lemma 1.1.6: For any two distinct lines ℓ1 and ℓ2 in a plane, either ℓ1||ℓ2 or ℓ1 and ℓ2 have exactly one point in common.

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Answer #1

Let the two lines 1 and 2 in slope intercept form be given by

The points of intersection of the two lines is / are given by solving above equations, to get

If m1 = m2, then the lines are parallel, as the slopes of two lines will be equal. If the slopes are not equal, then the points of intersection is given by equation (1), which is unique for non-parallel lines.

Hence, the theorem.

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