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If your definition of parallel lines is that they never meet, then the answer is yes. If, however, the definition is that they remain equidistant from one another at all points, then, in my opinion, the answer is no.

It is difficult to explain the second without recourse to diagrams which are very difficult to manage on this site. So consider a cube with vertices ABCD forming the top face and EFGH (in corresponding order) forming the bottom face.

Now AB is parallel to the bottom plane - EFGH. And AB is clearly parallel to EF and any line parallel to EF. But is AB parallel lines such as FG? True, they will never meet but the distance between them increases as you move away from BF - ie they are not the same distance apart.

Incidentally, Euclid's parallel postulate was phrased in a very different way from the one most mathematicians come across it. That version is a much later equivalent statement.

Q: Is it true if a line is parallel to plane it is parallel to any line in the plane?

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It is a transversal line

No. A line can be contained by many, many planes, Picture this, A rectangle with corners - going clockwise - A, B, C and D is the screen of your computer. This is a plane figure. 1 inch away from it a line runs from A1 to C1. The line is parallel to the plane. Now, take a sheet of paper with corners E, F, G and H, and place corner E at corner A of the screen, and place corner F at corner C of the screen. The Line AI is now 'contained' in the plane EFGH. and EFGH is perpendicular to ABCD.

Yes, they can. Since three points define a plane, take any two points on one line and a point on the other line, and form the plane with those three points. Once you have that, then use Euclid's test to see if they are parallel. Alternately, if the planes themselves are parallel, then the lines are as well, since they definitely will never intersect.

Yes, any points that are located on the same line will also be on the same plane. You can have more than one plane intersect a given line, but any points on that line will necessarily be on all the planes that intersect that line.

There cannot be parallel lines in any [plane] triangles.

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It is a transversal line

No. A line can be contained by many, many planes, Picture this, A rectangle with corners - going clockwise - A, B, C and D is the screen of your computer. This is a plane figure. 1 inch away from it a line runs from A1 to C1. The line is parallel to the plane. Now, take a sheet of paper with corners E, F, G and H, and place corner E at corner A of the screen, and place corner F at corner C of the screen. The Line AI is now 'contained' in the plane EFGH. and EFGH is perpendicular to ABCD.

They have the same angle relative to any plane or line and never intersect.

It is any plane parallel to the sagittal plane.

True.

Line segments are congruent if they have the same length. However, they need not be parallel. They can be at any angle or orientation on the plane

no, if they are both in the same plane and IF EXTENDED INDEFINITELY would never intersect at any point then the segments are considered parallel.

Yes, they can. Since three points define a plane, take any two points on one line and a point on the other line, and form the plane with those three points. Once you have that, then use Euclid's test to see if they are parallel. Alternately, if the planes themselves are parallel, then the lines are as well, since they definitely will never intersect.

Yes, since any line can be contained in a plane.

If the planes are non-intersecting, then they're parallel. Any line that intersects one of them intersects both of them.

Yes, any points that are located on the same line will also be on the same plane. You can have more than one plane intersect a given line, but any points on that line will necessarily be on all the planes that intersect that line.

There cannot be parallel lines in any [plane] triangles.