Space Geometry

Rules

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Parallel

1- How to prove (ABCD) // (EFGH) :


- (AB) // (EF) (AB) ∩ (BC)

- (BC) // (FG) (EF) ∩ (FG)


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2- (P) ∩ (Q)=(d)


-(∆) ⊂ (P)

-(∆') ⊂(Q)

=>(d)//(∆) // (∆')

Perpendicular

1- To prove (d) ⊥ (P)

We prove (d)⊥ to 2 intersecting lines in (P)


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Example:

Prove (AE) ⊥ (EFG)


(AE) ⊥ (EF) (AEFB is a rectangle)

(AE) ⊥ (EH) (AEHD is a rectangle)

=> (AE) ⊥ (EFH) = (EFG)

2- If a line is perpendicular to a plane then it is perpendicular to any line contained in this plane

Example:


Prove: (AE) ⊥ (EG)


since (AE) ⊥ (EFG)

and (EG) ⊂ (EFG)

=> (AE) ⊥ (EG)

Dihedral Angle

Angle between two planes

1) Choose the common line between (P) and (Q): (AB)

2) Select straight line in (P) ⊥ (AB)

3) Select a straight line in (Q) ⊥ (AB)

4) Find the angle using any method in geometry (trigonometry, triangles, circles..)


Angle between line and a plane

To find angle between (d) and (P):

1- Make the orthogonal projection from (d) to (P)

2- Find the angle using any method in geometry (trigonometry, triangles, circles..)

Mediator plane

Definition: Mediator plane of [AB] is a plane ⊥ to [AB] at its midpoint


Property: Any point on Mediator Plane is equidistant from the extremities.


You can prove a mediator plane by proving 3 points on Mediator Plane equidistant from extremities

Perpendicular planes

To prove two planes perpendicular:


prove a plane (P) perpendicular to a line (d) contained in (Q)

=> (P) ⊥ (Q)

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Parallel Planes

To prove two planes parallel:


Prove two intersecting straight lines parallel in (P) parallel to two intersecting straight lines in (Q)

Axis of Circle

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Done By:

Kareem Bayoun

Grade 11 "B"