Space

Orthogonality in space (Math Summary)

1.Orthogonal lines

-when two lines are orthogonal any paralel to one is orthogonal to the other.

-when two lines are paralel any orthogonal to one is orthogonal to the other.

Remark:two orhtogonal lines are not necessarily intersecting, but perpendicular lines are intersecting.

Line perpendicular to plane

A line is perpendicular to the plane if its orthogonal to all the lines in the plane.
Note:if a line doesnt intersect the plane its either parallel to the plane or belongs to it.

How to prove:

to prove that a line is perpendicular to a plane it is enough to show that it is orthogonal to two intersecting lines in the plane.

properties:

1.If two line are parallel any perpendicular plane to one is perpendicular to the other.

2.If two planes are parallel then every line perpendicular to one is perpendicular to the other.

3.If two line are nperpendicular to the same plane then they are parallel.

4.If two planes are perpendicular to the same line then they are parallel.

5.If a line is perpendicular to a plane then every line perpendicular to that line either belongs to the plane or is parallel to the plane.

6.If a line is perpendicular to the plane then every line parallel to this plane is orthogonal to the perpendicullar line.

Perpendicular planes

-Two planes are parallel if and only if one of them contains a straight line perpendicular to the other.

-If two planes are parallel then every line in one of them perpendicular to the intersection line is perpendicular to the other plane.

-If two planes are parallel then every plane perpendicular to one is perpendicular to the other.

-Through a line not perpendicular to a given plane we can draw one plane perpendicular to the given plane.

Mediator plane

-A mediator plane is a plane perpendicular to a segment at its midpoint.

-Set of points equidistant from two points.


Dihedral Angle

-The angle between two intersecting planes is called dihedral.

-To find the dihedral angle we need to find the common line (intersection line) two lines each from each plane perpendicular to the intersection line at the sam line and then find the angle.

Bisector of a dihedral plane

Defenition:Consider two planes (P,Q) a dihedral,(d) its edge and (B) a plane such that the plane angle of the dihedrals (P,B) and (B,Q) have the same measure.We say then (B) is the bisector of the dihedral (P,Q).

Property:The bisector (B) of the dihedral (P,Q) is the set of points equidistant from (P) and (Q)