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Is it true if a line is parallel to plane it is parallel to any line in the plane?
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.
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What is a line which intersects two parallel lines?
It is a transversal line
Is it true that if a line is parallel to a plane then any plane containing that line is always parallel to the given plane?
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.
Can lines on different planes be 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.
Is it true if points are collinear they are also coplanar?
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.
How many pairs of parallel line does a triangle have?
There cannot be parallel lines in any [plane] triangles.
What is a line which intersects two parallel lines?
It is a transversal line
Is it true that if a line is parallel to a plane then any plane containing that line is always parallel to the given plane?
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.
What is parallel lines properties?
They have the same angle relative to any plane or line and never intersect.
What is a parasagittal?
It is any plane parallel to the sagittal plane.
An equation for a line in the plane allows you to find the x- and y-coordinates of any point on that line?
True.
What is the definition of congruent line segment in geometry?
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
Do parallel line segments have to be the same length?
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.
Can lines on different planes be 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.
If four points are collinear they are also coplanar?
Yes, since any line can be contained in a plane.
How do you draw a line that intersects two nonintersecting planes?
If the planes are non-intersecting, then they're parallel. Any line that intersects one of them intersects both of them.
Is it true if points are collinear they are also coplanar?
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.
How many pairs of parallel line does a triangle have?
There cannot be parallel lines in any [plane] triangles.
