# Lines and planes - general concepts

I've come across a book that has this general questions about lines and planes. I can't agree with some of the answers it presents, for the reasons that I'll state below:

True or False:

• Three distinct points form a plane - BOOK ANSWER: True - MY ANSWER: False, they cannot belong to the same line

• Two intersecting lines form a plane - BOOK ANSWER: True - MY ANSWER: False, they can be parallel and coincident lines.

• Two lines that don't belong to a same plane are skew - BOOK ANSWER: True - MY ANSWER: True

• If three lines are parallel, there is a plane that contains them - BOOK ANSWER: True - MY ANSWER: False, they can be parallel and coincident lines.

• If three distinct lines are intersecting two by two, then they form only one plane - For this last one there's no answer and I'm not sure about the conclusion.

If you could help me, I appreciate it.

Thank you.

• It depends on what your definition of intersecting lines is. If they intersect at one point only, they cannot be parallel, and form a plane, if they are the same line, then obviously you are left with a line. If three lines are parallel they are by definition all in the same plane, if two lines are considered to be parallel, they cannot be coincident as they don't touch by definition. And I don't understand the meaning of the last statement. – BadAtMaths Sep 3 '15 at 15:30
• I think when the book says "two intersecting lines", it means "two distinct (non-coincident) intersecting lines". – 5xum Sep 3 '15 at 15:31
• I agree with you on the first point. The points cannot be on the same line. – Colm Bhandal Sep 3 '15 at 15:34

Assuming 3D Euclidean geometry:

• On point 1 if the three points are collinear there is more than one plane

• I would agree with the book and disagree with you on point 2 for a suitable definition of intersecting (at exactly one point).

• I would disagree with the book on point 4 (consider the three parallel edges of a triangular prism).

• I would say True for point 5, again for a suitable definition of intersecting two-by-two (the plane is defined by the three points of intersection, which do not lie on a single line and all three lines lie on it)

• Hi @Henry, I completely agree with you on your points, thank you. – bru1987 Sep 3 '15 at 15:41
• can you also say that two lines that don't belong to a plane are definitely skew? – bru1987 Sep 3 '15 at 15:42
• @bru1987 Two lines that don't belong in a plane are definitely skew, by number 2. – Morgan Rodgers Sep 3 '15 at 15:45
• @MorganRodgers ah I see, thank you. – bru1987 Sep 3 '15 at 15:46

(1) For three points to be on a plane, they need not be on the same line. Draw a triangle on a sheet of paper, and you will see what I mean.

(2) If they are coincident lines, they are not called 'intersecting' lines. Intersecting lines meet at one point ONLY.

(4) You are right. (However, even if they were all coincident they would still belong to some plane). But, in general they need not belong to a plane.

(5) True. Referring to (2), intersecting lines lie in a plane. If line A intersects line B, then A and B are in same plane. If B intersects with line C, then B and C are in same plane. Thus, intersection of A and C should also be in the same plane.

• As for point 5, can you also say that two lines that don't belong to a plane are definitely skew? – bru1987 Sep 3 '15 at 15:45
• For certain. Two distinct lines can either be parallel, intersecting or skew. If they are the first two, then they are in the same plane. If they are not in the same plane, the only option left is they are skew. – Mihir Sep 3 '15 at 15:48