Space

Math

Lines and Planes

Line perpendicular to a plane:

If a line is perpendicular to a plan, then it is perpendicular to any line in the plane.

To Prove a line (Δ) perpendicular to a plane (P):

To prove line (Δ) perpendicular to two lines in plane (P).


Lines Paralle to a plane:

To prove Line (Δ) Parallel to two lines in plane (P).


Angle of a straight and a Plane: Λ

The acvute angle of plane (P) and a straight line (Δ) is the angle HIA where H is the orthogonal projection of any point A of line (Δ) on plane (P)

Planes

Plane Perpendicular to a Plane:

To prove a plane (Q) prependicular to a plane (P) prove lines (Δ) and line (Δ') contained in (Q) perpendicular to one line (L) contained in plane (P).


Plane Parallel to a plane:

Method (1): To prove plane (P) parallel to a plane (Q), prove that lines (Δ) and line (Δ') contained in plane (P) are parallel to two lines (L) and (L') contained in (Q).

Method (2): To prove two planes (P) and (Q) are parellel, prove them are perpendicular to the same line.


Dihedral angle of two planes:

To find the angle between two planes:

Let line (Δ) be the line of intersection between planes (P) and (Q), the acute angle of the two planes (P) and (Q) is the acute angle of two perpendiculars draw through a point A of line (Δ) in each of the two planes.


Bisector Plane:

Consider the planes (Q), (R), (P) where (R) is the bisector plane and line (Δ) is the intersection line of thr three planes.

To prove (R) the bisector plane of Planes (P) and (Q), prove:

Method(1): Prove the angle between (P, R) is eaual to angle between (R, Q).

Method (2): Prove that the distance between (R,Q) is equal to the distance between (R,P) + the intersection line (Δ).


Mediator Plane:

To prove mediator plane :

Method (1): Prove the plane perpendicular to a segment (AB) at its midpoint K is the mediator plane of [AB].

Method (2): Prove that the three points of the mediator plane is equidistatn to segment [AB] point in the mediator plane is equidistant to segment [AB].


Axis of a plane:

Line (Δ) is the axis of a plane (P) when line (Δ) is perpendicular to plane (P) at its mid point.

Method (1): To prove axis of a plane: prove that any point on line (Δ) is equidistant to the plane (P):

Method (2): To prove axis of a plane, prove that (A), prove that AB=AC=AD, where (A) is a point on line (Δ), and B, C, D are points belonging to plane (P) + (AO) perpendicular to plane (P), where o is the center of plane (P).

Sapce Geometry

Leial Makari

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