Solve both parts and justify the conditions used. Part A: Find the dihedral angle between the coordinate planes x=0 and y=0. Part B: A line l is outside plane α and parallel to a line m in α. State and justify the conclusion.

01 / Foundation#Your turn

\begin{gathered}\text{B: }l\not\subseteq\alpha,\quad m\subseteq\alpha,\quad l\parallel m\end{gathered}

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

A: Use perpendicularity tests with intersecting directions, then identify the relevant projection or section.

Hint 2

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

Worked solution

Part A reasoning
Use perpendicularity tests with intersecting directions, then identify the relevant projection or section.
Calculate or simplify this relation.
$\mathbf n_1=(1,0,0),\quad\mathbf n_2=(0,1,0),\quad\mathbf n_1\cdot\mathbf n_2=0$
The section gives two perpendicular coordinate axes.
Part B reasoning
Apply a parallelism criterion with all its incidence and outside-plane conditions.
Calculate or simplify this relation.
$l\parallel m\subseteq\alpha,\ l\not\subseteq\alpha\Rightarrow l\parallel\alpha$
Without the outside-plane condition, l could be contained in α.

A: The requested value is 90. B: l is parallel to α.

Checks and common pitfalls: The section gives two perpendicular coordinate axes. Without the outside-plane condition, l could be contained in α.

Reasoning checklist · self / teacher assessment

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

Introduction to solid geometry: mixed assessment