# Space

### تعال نتعرف على قواعد ال"Space"

## Things To Learn :

## Parallel & perpendicular planesHow to prove : 2 planes parallel ,2 planes perpendicular, line perpendicular to a plane... | ## Anglesbetween two planes , between a line and a plane, angle bisector | ## Other rules and lawsMediator Plane ,Axis of a circle Cos , Sin & Area law |

## 1)Perpendicular planes

## 1If a line is perpendicular to each of two intersecting lines at their point of intersection, then the line is perpendicular to the plane determined by them. | ## 2If a line is perpendicular to a plane, then every plane containing the line is perpendicular to the given plane. | ## 3If a line (AB) is perpendicular to a plane (P) , then it is perpendicular to every line contained in the plane, ex: (CE) |

## 1

If a line is perpendicular to each of two intersecting lines at their point of intersection, then the line is perpendicular to the plane determined by them.

## 2

If a line is perpendicular to a plane, then every plane containing the line is perpendicular to the given plane.

## 2)Parallel Planes

## 12 Planes (A) and (B) are parallel if 2 intersecting lines in in (A) are parallel to 2 intersecting lines in (B) | ## 2If a plane intersects (q) two parallel planes(l) and (r), then the intersection is two parallel lines. (AB) parallel to (CD) | ## 3If two planes are perpendicular to the same line, they are parallel. |

## 1

2 Planes (A) and (B) are parallel if 2 intersecting lines in in (A) are parallel to 2 intersecting lines in (B)

## 2

If a plane intersects (q) two parallel planes(l) and (r), then the intersection is two parallel lines. (AB) parallel to (CD)

## "2oo3a" Note :

**If (d) is parallel to a plane (P) , then (d) isn't parallel to every line (t) contained in (P)**

## 3) Angles

## Line&PlaneTo find Angle between line and plane: Find the orthogonal projection of the line on the plane | ## 2 PlanesTo Find Dihedral Angle Between 2 Planes: 1) Find the common line (a) 2) Select a strainght line (q) & (p) from each plane , perpendicular to (a) | ## BisectorTo prove plane(P) bisects 2 intersecting planes(C)&(D): 1) Prove angle POF between (P) & (D) = angle POE between (C) & (P) 2)OR : Prove 3 points on (P) equidistant from (C) & (D) |

## Line&Plane

To find Angle between line and plane: Find the orthogonal projection of the line on the plane

## 2 Planes

To Find Dihedral Angle Between 2 Planes: 1) Find the common line (a) 2) Select a strainght line (q) & (p) from each plane , perpendicular to (a)

## 4)Other Rules

## Mediator PlaneMediator plane of [AB] : plane perpendicular to [AB] at its midpoint. -Any point on it is equidistant from the 2 extremitiesA&B -To prove Mediator Plane :You can prove 3 points on it equidistant from the extremities | ## Axis of a circle(OR) is Axis of circle (C) at O: -Any point on (OR) is equidistant from any point on (C). -Prove : -By defenition -RI=RK=RA & (RO) perpendicular to (C) at its center O | ## VIP RulesCos rule : a^2 = b^2 + c^2 -2bc cos(A) ___Sin rule : a/sin(A) = b/sin(B) = c/sin(C) ___Area rule : A = 0.5ab x sin (C) |

## Mediator Plane

Mediator plane of [AB] : plane perpendicular to [AB] at its midpoint. -Any point on it is equidistant from the 2 extremitiesA&B -To prove Mediator Plane :You can prove 3 points on it equidistant from the extremities

## Axis of a circle

(OR) is Axis of circle (C) at O: -Any point on (OR) is equidistant from any point on (C). -Prove : -By defenition -RI=RK=RA & (RO) perpendicular to (C) at its center O

## "2oo3a tinsa" Note:

**1)Diagonals in a cube = Side radical 3 , but they aren't parallel.**

**2)In locus : A circle of diameter [AB] becomes a sphere in space .However it's intersection with a plane gives a circle.**

## Did you know ?

**arose from practical problems faced by the antients , especially Egypcians around 2000 B.C**

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