Question

Prove (with neutral geometry) If the angle of parallelism for a point P and a line...

Prove (with neutral geometry)
If the angle of parallelism for a point P and a line RS is Less Than 90 degrees, then there exist At Least Two lines through P that are parallel to line RS.

Please explain step by step and include a diagram. Thanks so much!
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Answer #1

The meaning of the angle of parallelism for a point P and a line RS is less than 90 degrees is that there is a line through P (say PT) which makes an angle less than 90 degrees with the perpendicular from P to RS (say PQ) and PT is parallel to RS (that is they don't intersect)

Now construct a perpendicular to PQ at P(Call it PU). Constructing perpendiculars is possible in neutral geometry.

Since PQ is perpendicular to both PU and RS, they have to be parallel. PU is not equal to PT as PT makes an angle less than 90 degrees with PQ. Hence they are different.

Therefore there are two parallel lines through P to RS

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