# I need to prove that every line l is contained by at least 2 planes

• LCharette
I-1 Each two points determines a line)In summary, we can prove that every line is contained by at least two planes by using only the incidence axioms and the fact that each line has at least two points and space has at least three noncollinear points. By creating two planes that contain the line and a third noncollinear point, we can show that the line is contained by at least two planes.
LCharette

## Homework Statement

Use only incidence axioms to prove that every line is contained by at least two planes.

## The Attempt at a Solution

1. Let l be any line (Given)
2. l has at least two points A and B such that l=AB (I-5(4) Each line has at least two points)
3. Space has at least 3 noncollinear points, that is A,B, and C (I-5(2) Space has at least three noncollinear points)
4. There exists lines AC and BC (I-1 Each two points determines a line)

From here I do not know where to go or even if I am on the right track.

5. Since AC and BC are determined by the points A and B, they must also contain the point C (I-3 If a point lies on a line, then it lies on all planes that contain the line)
6. Therefore, l=AB is contained by the planes ABC and BAC (I-2 If two distinct lines intersect, then they intersect in exactly one point.)
7. Since l=AB is contained by two planes, every line is contained by at least two planes.

## 1. How do you prove that every line is contained by at least 2 planes?

To prove that every line is contained by at least 2 planes, you must use the definition of a plane, which states that a plane is a flat, two-dimensional surface that extends infinitely in all directions. From this definition, it follows that every line must be contained by at least 2 planes, as a line can be thought of as the intersection of 2 planes.

## 2. What is the significance of proving this statement?

Proving that every line is contained by at least 2 planes is important in mathematics and physics, as it is a fundamental concept in geometry and spatial reasoning. It also has practical applications in fields such as engineering and computer graphics.

## 3. Can you provide an example to illustrate this statement?

Imagine a pencil lying on a flat table. The pencil can be thought of as a line, and the table can be thought of as a plane. However, the pencil is also contained by the vertical plane created by the table's surface and the horizontal plane created by the pencil itself. This example shows that every line is indeed contained by at least 2 planes.

## 4. Is it possible for a line to be contained by more than 2 planes?

Yes, it is possible for a line to be contained by more than 2 planes. For example, a line in three-dimensional space can be contained by an infinite number of planes, as long as those planes do not intersect each other.

## 5. How does this statement relate to other geometric concepts?

The concept that every line is contained by at least 2 planes is closely related to the concept of parallel lines and planes. In order for two lines to be parallel, they must lie in the same plane. And if two planes are parallel, they will never intersect and will therefore contain the same lines. This connection helps to further reinforce the idea that a line must be contained by at least 2 planes.

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