# SPACE

## 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.

## 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

(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)