Unizor - Geometry3D - Lines and Planes - Problems 5

Unizor - Creative Minds through Art of Mathematics - Math4Teens

Let us recall the main axioms of solid geometry we will be using.

Axiom 1. If two points of a straight line belong to a plane, every point of this line belongs to this plane.

Axiom 2. If two planes have a common point, they intersect along a straight line passing through this point.

Axiom 3. For any three points not lying on the same straight line there is one and only one plane that contains them.

Now we are ready to solve problems.

1. Construct a plane that intersects three different planes.

2. Prove that two angles in three-dimensional space with correspondingly parallel sides, one formed by lines a and b intersecting at point P and another formed by lines a' and b' intersecting at point P', are congruent.
P = a∩b; P' = a'∩b';
a ∥ a'; b ∥ b'
⇒ ∠aPb = a'P'b'

3. Prove that a line a parallel to another line b, that is perpendicular to a plane γ, is itself perpendicular to the same plane.
a ∥ b; b⊥γ
⇒ a⊥γ

4. Prove that two perpendiculars a and b to the same plane γ are parallel to each other.
a⊥γ; b⊥γ
⇒ a ∥ b

5. Given a plane γ and a straight line a parallel to it.
Prove that the distance from any two points M and N on a line a to a plane γ is the same.
a ∥ γ; M∈a; N∈a;
P∈γ; MP⊥γ;
Q∈γ; NQ⊥γ
⇒ MP=NQ

6. Prove that two planes γ and δ, parallel to another plane ρ, are parallel to each other.
γ ∥ ρ; δ ∥ ρ
⇒ γ ∥ δ