LEARN · EXPLAIN · REVISE

Does a line parallel to a plane have to be parallel to every line in that plane? Give a counterexample.

Read the idea, work independently, then explain what changed.

高一必修第二册(A版).pdf · 8.5 · PDF 140 / printed page 133

TOPIC 01

Parallel lines and planes in space

Build understanding of parallel lines and planes in space through definitions, contrasting cases and justified applications.

Use definitions, valid spatial reasoning and metric checks for parallel lines and planes in space.
Connect representations and justify the steps, including boundary cases.
Explain a solution and apply the idea to a changed situation.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Transfer#Your turn

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Compare direction vectors.

Hint 2

Plane parallelism constrains a normal component, not one unique direction.

Worked solution

Compare direction vectors.
Calculate or simplify this relation.
$l\text{ direction }(1,0,0);\quad m\text{ direction }(0,1,0)$
There are many different directions in a plane.

No: l=(t,0,1) is parallel to z=0, but not parallel to the y-axis in that plane.

Checks and common pitfalls: There are many different directions in a plane.

Reasoning checklist · self / teacher assessment

Think first. Reveal a hint when the class is ready.

Teacher preparation and assessment

What must be true before using the main rule for parallel lines and planes in space?
Which representation makes this task easier, and why?
Change one assumption. Does the conclusion survive?

Use definitions, valid spatial reasoning and metric checks for parallel lines and planes in space.
Line-plane criterion: An outside line parallel to a line in a plane is parallel to the plane.
Plane-plane criterion: Two intersecting directions are needed to establish a plane orientation.
Close with: conditions → representation → reasoning → check.

Expected reasoning: An outside line parallel to a line in a plane is parallel to the plane.
Expected reasoning: Two intersecting directions are needed to establish a plane orientation.
Expected correction: Parallel to the same plane does not mean parallel to each other.