g10-14-03 · Additional mathematics
Two test lines must intersect
Do two parallel test lines prove that two planes are parallel?
1 MIN · PREDICT ON PAPER
Tilt an upper plane about one horizontal direction. Predict whether two parallel horizontal lines detect the tilt.
Discuss or write first. No typed answer is required to continue.
Two independent directions and distinct planes are required.
2 MIN · EXPLAIN AND REVISE
Which condition does your claim need?
Compare at least two cases. Write a reason, then check the explanation together.
Reveal explanation and conditions
Two intersecting lines in one plane, each parallel to the other plane, determine two independent directions. Two parallel test lines determine only one direction and can miss a tilt.
Conditions: The planes must be distinct. Coincident planes are not strictly parallel in the no-intersection convention used here.
Mathematical justification: For z=h+sx x+sy y, both independent direction vectors(1,0,sx),(0,1,sy) are horizontal exactly when sx=sy=0. A nonzero h then separates the planes.
1 MIN · EXIT CHECK
Must the two lines in a plane-plane parallelism test intersect?
Check after answering
Yes, to supply two independent directions; also exclude coincident planes.
Teacher guidance and extension
Before: Points, lines, planes and vector directions.
Ask: Which direction is never checked if both test lines run the same way?
Watch for: Replacing “two intersecting lines” with any two lines.
Challenge: Use z=y+1 to produce a counterexample with two horizontal parallel lines.
Suggested timing: predict 1 min, compare 3 min, explain 2 min, exit check 1 min. Observations can suggest a claim; they do not prove it for every input.
Optional local notebook
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