1.5  Parallel Lines & Planes

Lines and planes are classified by whether or not they intersect and how they intersect.  At this time we are only going to consider what happens when lines and planes do not intersect.

Planes:

When it comes to planes there are really only two possibilities, either they intersect or they don't.  If two planes do not intersect, then they are parallel.

Example A

Problem: Given the following figure, can you identify all parallel planes?

Solution: If we think of this figures as a simple box, we would realize that the opposite sides would be parallel.  S we would think of the sides of the box as planes. so would have: Here we can see the green sides are on opposite sides of the box and so the these 2-planes would be parallel. Here we can see that the red top and the orange bottom are on opposite sides of the box and so the these 2-planes would be parallel. Here we can see that the purple front and the blue back are on opposite sides of the box and so the these 2-planes would be parallel.

Watch the video:

So when it comes to planes they will either intersect ( which will always be a line ) or they will never intersect ( like the floor and ceiling of a room ).

Lines:

There are three possible ways for a pair of lines to be related to each other..

• Coplanar Lines, if two lines are on the same plane and do not intersect, then the lines are parallel.
  
• Coplanar Lines, if two lines are on the same plane and intersect ( at a point ), then the lines are not parallel.
  
• Noncoplanar Lines, if two lines are not on the same plane and do not intersect, then are lines are skew.

Notation:  Rather than always use the word "Parallel" we can use the symbol: " // " to indicate parallel lines.

We have already looked at what happens when we have two lines that intersect, now what happens if two lines do not intersect.

Example B

Problem: Given the following figure, can you identify all parallel lines?

Solution: If we can identify all of the lines that are on the same plane we would then know which lines are parallel. Here we see that both the purple and blue lines are on the yellow plane and since they are on the same plane and do not intersect these lines are parallel. These lines seem to be going in the same direction, but are they parallel? Since we can identify that we have two lines on each plane, we would be able to state that we have parallel lines.  Would the red and blues or the purple and green lines be parallel? Look at this figure Could the blue and green lines be parallel?  Could we make these two lines to be on the same plane? Here we see that both of the lines are on the same plane, so yes these lines are parallel

Watch the video:

Example C

Problem: Given the following figure, can you identify all skew lines?

Solution: Lines that are skew to each other will not intersect and will not be on the same plane. If we look at the red and blue lines they are not on the same plane and do not intersect. However if we were  view the figure from the right side, we would see: So the lines do not intersect and are not on the same plane, so they are skew

Watch the video:

Now you do...

Use Geogebra Cube App below to create a pair of lines that are :

• Parallel
   
• Skew
   
• Intersecting

Practice Problems

• Use the Geogebra Cube App to create the following lines
   
• Identify if the lines are either parallel. skew or intersecting
   
• Take a screenshot of the figure and paste into the Jamboard
  for this assignment

A.  Line:  AB     Line:  CG
      Are the lines: Parallel " // ". Skew or Intersecting?

B.  Line:  AG     Line:  CE
      Are the lines: Parallel " // ". Skew or Intersecting?

C.   Line:  HB     Line:  FG
      Are the lines: Parallel " // ". Skew or Intersecting?

D.  Use the POLYGON tool to shade in a pair of parallel sides.

Assignment:

• Use the Geogebra Cube App to create the following lines
   
• Identify if the lines are either parallel. skew or intersecting
   
• Take a screenshot of the figure and paste into the Jamboard
  for this assignment

1.  Line:  CB     Line:  HG
      Are the lines: Parallel " // ". Skew or Intersecting?

2.  Line:  CD     Line:  FG
      Are the lines: Parallel " // ". Skew or Intersecting?

3.  Line:  DE     Line:  FB
      Are the lines: Parallel " // ". Skew or Intersecting?

4.  Line:  EB     Line:  HA
      Are the lines: Parallel " // ". Skew or Intersecting?

5.  Line:  CB     Line:  HD
      Are the lines: Parallel " // ". Skew or Intersecting?

6..  Line:  CH     Line:  AF
      Are the lines: Parallel " // ". Skew or Intersecting?

7..  Line:  AD     Line:  HC
      Are the lines: Parallel " // ". Skew or Intersecting?

8..  Line:  CB     Line:  HG
      Are the lines: Parallel " // ". Skew or Intersecting?

9..  Line:  CA     Line:  HF
      Are the lines: Parallel " // ". Skew or Intersecting?

10.  Use the POLYGON tool to shade in 3 intersecting sides.
        Note: shade each side a different color

11.  Can 3-planes intersect at only one point?