Parallelism of a line and a plane, and of two planes
Answers are for parents and teachers.
1
In triangle ABC, B₁ ∈ AB, C₁ ∈ AC and B₁C₁ ∥ BC; B₁C₁ = 8 and BC = 20. Find the ratio AB₁ : B₁B.
AB₁ : AB = 8 : 20 = 2 : 5, so AB₁ : B₁B = 2 : 3.
2
In a cube, to which face plane is line B₁D₁ parallel? Justify.
To plane ABCD: B₁D₁ ∥ BD, BD lies in that plane and B₁D₁ does not. Of the other faces, one contains B₁D₁ and the four side faces meet it at B₁ or D₁.
3
Three parallel planes cut segments 5 and 9 from one line and x and 18 from another. Find x.
10
4
Briefly justify that opposite faces of a parallelepiped lie in parallel planes.
Two intersecting edges of one face are parallel to two edges of the opposite face; this is what the criterion for parallel planes requires.
5
If a ∥ α, how many lines of α are parallel to a? Explain.
Infinitely many: every line of α in the direction of a (take one such line and draw parallels to it).