14 Parallelism and perpendicularity in space · g10-14-01 · 5–10 min
Not intersecting does not imply parallel
Can two space lines neither intersect nor run parallel?
Predict: Compare AB with an upper parallel edge,an adjacent edge and a noncoplanar vertical edge.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
| Case 1 | |
| Case 2 | |
| Case 3 | |
Revise: My claim needs the condition… because…
Exit: Does “two lines do not meet” prove they are parallel in space?
14 Parallelism and perpendicularity in space · g10-14-02 · 5–10 min
A parallel line must lie outside the plane
Can the outside-plane condition be omitted?
Predict: Keep a line direction parallel to the floor and lower its height to0. Predict its relation to the floor.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
Case 1
Starting height h = 1; Vertical direction k = 0 | |
Case 2
Starting height h = 0; Vertical direction k = 0 | |
Case 3
Starting height h = 1; Vertical direction k = -1 | |
Revise: My claim needs the condition… because…
Exit: A line with points(t,0,1) has what relation to z=0?
14 Parallelism and perpendicularity in space · g10-14-03 · 5–10 min
Two test lines must intersect
Do two parallel test lines prove that two planes are parallel?
Predict: Tilt an upper plane about one horizontal direction. Predict whether two parallel horizontal lines detect the tilt.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
Case 1
Plane offset h = 1; Slope in x = 0; Slope in y = 0 | |
Case 2
Plane offset h = 1; Slope in x = 0; Slope in y = 1 | |
Case 3
Plane offset h = 0; Slope in x = 0; Slope in y = 0 | |
Revise: My claim needs the condition… because…
Exit: Must the two lines in a plane-plane parallelism test intersect?
14 Parallelism and perpendicularity in space · g10-14-04 · 5–10 min
Perpendicular to two parallel lines is not enough
Why must the two lines in the perpendicularity test intersect?
Predict: In the floor plane, draw two parallel x-directed lines and a y-directed line perpendicular to both. Is it perpendicular to the floor?
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
Case 1
Direction x = 0; Direction y = 0; Direction z = 1 | |
Case 2
Direction x = 0; Direction y = 1; Direction z = 0 | |
Case 3
Direction x = 0; Direction y = 0; Direction z = 0 | |
Revise: My claim needs the condition… because…
Exit: Which is sufficient: two parallel lines or two intersecting lines in the plane?
14 Parallelism and perpendicularity in space · g10-14-05 · 5–10 min
Perpendicular planes do not make every line normal
If two planes are perpendicular, is every line of one perpendicular to the other?
Predict: Fold a page to90°. Compare the crease direction with a line across the upright page.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
Case 1
Fold angle (degrees) = 90; Line direction within page (degrees) = 90 | |
Case 2
Fold angle (degrees) = 90; Line direction within page (degrees) = 0 | |
Case 3
Fold angle (degrees) = 45; Line direction within page (degrees) = 90 | |
Revise: My claim needs the condition… because…
Exit: In two perpendicular planes, is their intersection line perpendicular to either plane?
