Solve both parts and justify the conditions used. Part A: State a sufficient criterion for two distinct planes to be parallel using lines in one plane. Part B: How many planes contain two distinct parallel lines? Explain.

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Solve both parts and justify the conditions used. Part A: State a sufficient criterion for two distinct planes to be parallel using lines in one plane. Part B: How many planes contain two distinct parallel lines? Explain.

Use the domain, units and sampling assumptions stated in the question.
Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

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

Hint 2

B: Use precise incidence conditions; diagrams alone do not prove spatial relationships.

Worked solution

Part A reasoning
Apply a parallelism criterion with all its incidence and outside-plane conditions.
Calculate or simplify this relation.
$l,m\subseteq\alpha,\ l\cap m\ne\varnothing,\ l\parallel\beta,\ m\parallel\beta\Rightarrow\alpha\parallel\beta$
Two independent directions constrain the plane orientation.
Part B reasoning
Use precise incidence conditions; diagrams alone do not prove spatial relationships.
Calculate or simplify this relation.
$l\parallel m,\ l\ne m\Rightarrow\exists!\alpha:l,m\subseteq\alpha$
The three selected points are noncollinear and determine a unique plane.

A: Two intersecting lines in one plane must each be parallel to the other plane. B: The requested value is 1.

Checks and common pitfalls: Two independent directions constrain the plane orientation. The three selected points are noncollinear and determine a unique plane.

Reasoning checklist · self / teacher assessment

Solve part A with its stated restrictions.
Solve part B using an appropriate representation.
Give the reasoning and check conditions in both parts.

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

Solve both parts and justify the conditions used. Part A: State a sufficient criterion for two distinct planes to be parallel using lines in one plane. Part B: How many planes contain two distinct parallel lines? Explain.

Introduction to solid geometry: mixed assessment

What you will be able to explain

Try it. Leave your reasoning visible.

Solve both parts and justify the conditions used. Part A: State a sufficient criterion for two distinct planes to be parallel using lines in one plane. Part B: How many planes contain two distinct parallel lines? Explain.

Question sequence

Board plan

Anticipated thinking

Assessment checklist