The exercise involves analyzing a rectangular prism with labeled vertices and understanding the relative positions of various lines and planes. Here's a step-by-step breakdown to address the questions:

1) Sans justifier, donner les positions relatives des droites ou plans

(Without justification, give the relative positions of the lines or planes and intersections where applicable.)

• a) $(DGK)$ et $(EIB)$

  • These two planes do not intersect. They are in different regions of the solid.

• b) $(AFG)$ et $(EHD)$

  • These planes are parallel because they are two opposite faces of the rectangular prism.

• c) $(FCG)$ et $(EB)$

  • The plane $(FCG)$ contains the line $(EB)$. Hence, the line and the plane are secants.

• d) $(GHA)$ et $(KI)$

  • The plane $(GHA)$ and the line $(KI)$ are parallel and do not intersect.

2) Dire si les deux droites sont sécantes ou non en justifiant.

• a) Les droites $(AF)$ et $(DH)$

  • These lines do not intersect because they are skew lines (they are neither parallel nor in the same plane).

• b) Les droites $(DK)$ et $(CG)$

  • These lines are parallel. They belong to opposite faces of the prism and maintain constant distance.

Final Notes

This problem tests your understanding of the geometric relationships among lines and planes, particularly in a rectangular prism. Terms such as "parallel," "skew," and "secant" are crucial here.

Let me know if you would like further explanations, clarifications, or visual aids for these solutions!

Related Questions to Explore:

1. What is the difference between skew lines and parallel lines?
2. How do you determine if a line and a plane are secant?
3. What are the conditions for two planes to be parallel?
4. Can two skew lines become parallel under any transformation?
5. How do you compute the intersection point between a line and a plane?

Tip: Visualizing 3D figures on graph paper or with software (like GeoGebra) can make it easier to identify relationships between lines and planes.