LEARN · EXPLAIN · REVISE

Why are two parallel lines in α, both parallel to β, insufficient to prove α∥β?

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 / Standard#Your turn

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Apply a parallelism criterion with all its incidence and outside-plane conditions.

Hint 2

Construct tilted and horizontal planes sharing the x-direction.

Worked solution

Apply a parallelism criterion with all its incidence and outside-plane conditions.
Calculate or simplify this relation.
$\alpha:z=y,\quad\beta:z=0;\quad l=(t,1,1),\ m=(t,2,2)$
Both lines avoid β but their plane intersects β along y=z=0.

They supply only one direction; α can tilt about that direction and intersect β.

Checks and common pitfalls: Both lines avoid β but their plane intersects β along y=z=0.

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.