SPACE

Malaka Shoukair

Here are the rules that should be followed:

A. Lines and planes

1. If two straight lines (d) and (d') of space are perpendicular to the same plane then they are parallel to each other.


2. If a plane is perpendicular to a straight line (d) then all straight lines passing through a point B of the plane and perpendicular to (d) lie completely in the plane.


3. The acute angle of a plane (P) and a straight line (d) is the angle where H is the orthogonal projection of any point A of (d) on (P).


4. If a straight line (d) and plane (P) are perpendicular to the same line then they are parallel.


5. A straight line (d) is perpendicular to a plane (P) if and only if it is orthogonal to two intersecting lines of this plane.

( if a line(d) is perpendicular to a (P), then it is perpendicular to every line in (P))


6. if (l) perpendicular (d) and (d) contained in (Q)=> (l)perpendicular (Q)



when (l) perpendicular (Q)=> 1.(l) perpendicular (l') if (l') contained in (Q)

2.(P) perpendicular (Q) if (l) contained in (P)


Common perpendicular

a common perpendicular is a line perpendicular to 2 lines not necessarily to be in the same plane.

B. Planes

1. Angle between two planes

Finding the angle between planes (P) and (Q)

*You have to find:

1. A common line between planes (P) and (Q)
2. A line in (P) perpendicular to the common line

3. A line in (Q) perpendicular to the common line


(the angle here is a)

2. Prove Two planes parallel

Prove (P) and (Q) perpendicular to the same line (DC)

3. Prove Two planes perpendicular

To prove (P) perpendicular (Q):


Prove (l) contained in (Q) + (l) perpendicular (P)

C. Mediator plane of [AB]

Proof:

1. Definition: a plane perpendicular to a segment [AB] at its midpoint.


2.Proving 3 non-collinear points of a plane are equidistant to segment [AB]

D. Axis

Property: Any point on (d) is equidistant from any point on (C)

Proof axis:

1. Definition: (d) is axis of circle if (d) perpendicular (C) at O (center of (C))

2. A is a point on (d) and B,C,D are point on the circle
prove: AB=AC=AD and

(AK) perpendicular (C) =>K center (AK) axis

E. Find the anglular bisector between 2 planes

Property: Any point on [DB] is equidistant from the sides DA=DC


Proof:

1. Any point on R is equidistant from (P),(Q) and the intersection line.
2.The angle between (P,R)= angle between (Q,R).



REMARK:

1. If a straight line (d) //(P) it doesn’t mean that (d) // to any line in (P)

2. Perpendicular( intersecting)

orthogonal (skew, not intersecting)