Unizor - Geometry3D - Lines and Planes - Two Parallel and Transversal Pl...

Unizor - Creative Minds through Art of Mathematics - Math4Teens

We are considering three planes σ ∥ τ and γ that intersects both of them, correspondingly, at lines s and t.

Imagine that parallel planes σ and τ are positioned horizontally relatively to an observer, σ is above τ, and the intersecting plane γ is slanted in a way that an observer sees it from a side (so, all planes look like lines for an observer, two horizontal parallel to each other and one intersecting both of them).

Line s divides plane σ into left and right half-planes (we will use suffixes L and R).
Line t divides plane τ into left and right half-planes (we will use suffixes L and R).
Each of these lines, s and t, divide plane γ into up and down half-planes (we will use both subdivisions and suffixes U for up and D for down, which line s or t is used as an edge will be clear from the context).

Let's introduce some terminology similar to the one in plane geometry when we considered two parallel lines and another intersecting both of them.

Parallel Planes:
These are planes σ and τ.

Transversal Plane:
This is plane γ.

Vertical Dihedral Angles:
∠σLsγU and ∠σRsγD
∠σRsγU and ∠σLsγD
∠τLtγU and ∠τRtγD
∠τRtγU and ∠τLtγD

Corresponding Dihedral Angles:
∠σLsγU and ∠τLtγU
∠σRsγU and ∠τRtγU
∠σLsγD and ∠τLtγD
∠σRsγD and ∠τRtγD


Alternate Interior Dihedral Angles:
∠σLsγD and ∠τRtγU
∠σRsγD and ∠τLtγU

Alternate Exterior Dihedral Angles:
∠σLsγU and ∠τRtγD
∠σRsγU and ∠τLtγD

Consecutive Interior Dihedral Angles:
∠σLsγD and ∠τLtγU
∠σRsγD and ∠τRtγU

Mini Theorem

Vertical dihedral angles are congruent.
Corresponding dihedral angles are congruent.
Alternate Interior dihedral angles are congruent.
Alternate Exterior dihedral angles are congruent.
Consecutive Interior dihedral angles are supplementary.

Proof

Draw a plane δ perpendicular to edge s. It will be perpendicular to edge t as well because s ∥ t.
All the dihedral angles now can be represented by corresponding linear angles. Since the above statements are true for linear angles, they are true for dihedral.