# summary

### space

## Welcome To Our Massive World Of Space !!!

**Refresh Your Memory:**

- We need
points to form a plane.__3 NON-COLLINEAR__ __Co-planer__: belong to the same plane (can be intersecting-parallel or confoumded lines.)- planes may intersect at a line or a point.
- Cavaliers Perspective : (
**way to draw 3D on a paper**(the non-frontal faces ))

- Right angles aren't shown right - equal sides aren't shown equal (but assumed so when solving).
- Parallelism and Midpoints are preserved.
- Faces that are hidden are drawn dotted
**EVERYTHING**is preserved in a__frontal face__. (**ex**__:__a rectangle is shown as a rectangle not as a parallelogram.)

## Parallelism and planes :

__Parallel lines and planes:__- Every line parallel to a line contained in a plane is parallel to this plane or contained in it.
- If a line (d) is parallel a plane (P), every plane passing through (d)and cutting (P), cuts it along a line parallel to (d).
- Every line parallel to 2 intersecting planes is parallel to their line of intersection.
- If 2 intersecting planes pass through 2 parallel straight lines then their line of intersection is parallel to these to lines.

## Parallel Planes:

- If two planes are parallel, every line in one of them is parallel to the other
__plane.__ - If a plane (P) contains two intersecting lines parallel to another plane (Q)
parallel two another 2 intersecting lines in (Q) , then (P) is parallel to (Q).__or__ - (P) // (Q) and (P) // (R) then, (Q) // (R).
- If (P) // (Q) and (Q) perpendicular to (R), then (Q) perpendicular to (R).
- If 2 planes are parallel every plane which cuts one of them cuts the other and their intersections are parallel.

## If a straight line (d)//(P) DOESN'T MEAN (d)//(d') any line in (P)

## Perpendicular and Planes:

__BUT WE CAN'T GO BACKWARDS !!__## Angle between a plane and a line:

Draw an

**orthogonal line**(CF)**from the line**(CH)**to the plane**(FGHE),THEN **FHC **is the **angle** between (CH) and (FGHE)

## Mediator plane:

**It is a plane perpendicular to a line at its midpoint.**

__DEFINITION:____Property:__ Any point on the M.P is equidistant from the extremities of the segment .

__To Prove : __

- By Definition.
- By proving 3 non-collinear points of M.P equidistant from the extremities of the segment.

## Angular Bisector plane :

__A plane who bisects an angle between 2 planes/lines.__

**Definition:**** Properties:** Any pt. M on the plane is equidistant from the 2 sides of the angle.

**How to prove:**

- prove that
**angle between (P) and (R) = angle between (R) and (Q)** - prove that
**any point**on (R) is**equidistant**from (p) and (Q) ===> then the**bisector plane**would be this**point and the intersection line of the 2 planes.**

## Axis of the circle circumscribed:

**It is the**

__Definition:__**axis of the circle/plane**where it is

**perpendicular**to it at is

**center.**

__ Property:__ Any point on the axis is equidistant from any pt on the circle/plane.

**How to prove:**

- By definition
- A pt. on the axis equidistant to 3 pts on the circle/plane + perpendicular at its center.

## Some important rules:

**Cos Rule:**a^2=b^2+c^2-2bc.cosA**Sin Rule:**a/sinA = b/sinB= c/sinC**Area Rule:**A=1/2.absinC**Locus**is the place where a**variable point moves**.. In space it is mainly**a circle on plane (P)**determined in different ways.- All
**Previous Geometric Rules**areImportant!!!__VERY__