Spatial Reasoning

Two lines are coplanar: Sometimes

A B C D N This is an example of when two lines are coplanar, or on the same plane. Line AB and line CD are both contained in plane N.

In this example, the two lines are not coplanar. Line AB is contained in plane N, and line CD is contained in plane M. In order for these two lines to be coplanar, they would have to be on the same plane. B A C DN M

A line intersects a plane in one point: Always

A B N This example represents a line (AB) intersecting a plane (N) in one point (B). Because a line is an unbroken set of points that continues forever, it can only intersect a plane in one point.

Two planes intersect in a line: Always

A B NM In this example, two planes are intersecting in one line. Plane M, on the left, and plane N, on the right, come together at the line AB.

Two planes intersect in a single point: Never

A B M N Two planes cannot intersect in just one single point. A plane continues forever in all directions, and lines continue forever in two directions. Two planes would then have to intersect in a line, like shown above.

Planes have an edge: Never

Planes are defined as surfaces with length, width, but no thickness. A planes also is composed of lines, that continue forever. This means that planes continue forever. Although two planes may intersect in a line, like shown above, the planes N and M do not have an edge. NM A B

Two points are collinear: Always

A B In this example, two sets of points are shown. They are both contained in the same line as at least one other point, and this makes them collinear. A and B are on the same line. X and Y are on the same line. X Y

Three points are collinear: Sometimes

A B C In this example, three points are being shown as collinear, meaning they are located on the same line. Points A, B, and C are those collinear points. But three points will not always be collinear.

A B C This example shows three points that are not collinear. Points A and B can be on the same line, but point C is not on that same line. You could say point A and C are on the same line, but then point B is left out. So you can see how in this example only two of the points will be collinear.

Three points are coplanar: Always

A C B N In this example, the three points (A, B, and C) are coplanar. They are all on plane N. A plane continues on forever in all directions, and so the points connect to one another and then continue in lines that go on forever in two directions.

Four points are coplanar: Sometimes

A B C D N This example shows how four points can be coplanar. Points A, B, C, and D are all contained in plane N. This means they are all on the same plane, and that makes them coplanar.

This is an example of how four points can not be coplanar. Points A, B, and C are all contained in plane N, but point D is in plane M. This means that those four points are not all coplanar. A B C D N M

A point is a small, filled circle: Never

A B C X Y Z A point is a location, and does not have size, depth, or thickness. Small circles are sometimes used to represent where a point is because it is sometimes hard to explain where a point is. So in the example above, A, B, C, X, Y, and Z are different locations in the garden. The small circles help label them.