Space

Summary

Parallel Planes and Lines

IF:

*(P) is parallel to (Q)

and

(Q) is parallel to (R)

then

(P) is parallel to (R)


*(P) is parallel to (Q)

and

the intersection of (P) and (R) is (d)

then

the intersection of (R) and (Q) is (d') which is parallel to (d)



How to prove that a straight line is parallel to a plane?

(d) parallel to (d') + (l) parallel to (l') + (d) and (l), and (d') and (l') intersect.

Two planes intersect at a straight line

Perpendicular Planes and Lines

How to prove that a straight line is perpendicular to a plane?

Prove (d) perpendicular to 2 intersecting lines in (P)
IF:

(d) perpendicular to (P)

then

(d) is perpendicular to any line in (P)

IF:

(P) is parallel to (Q)
and

(Q) is perpendicular to (R)

then

(P) is perpendicular to (R)


How to prove two Planes perpendicular?

Shapes

angle between a line and a plane

How to find?

1) Find the orthogonal projection of point not in the plane on it.

2) Find angle between these 2 lines.

Dihedral Angle

A dihedral angle is the angle between two planes.

How to find?

1) Find the common straight line.

example: (BC)

2) Select a straight line in the first plane perpendicular to the fist to the common line.

example: (SB)

3) Similarly select one in the second plane.

example: (AB)

The angle is SBA

Mediator Plane

It's a plane perpendicular to a line at its midpoint.

Property

Any point of the mediator plane is equidistant from the extremities (MA=MB)

How to prove?

Methods:

a) Prove 3 non-cillinear points on the plane equidistant from the extremities.

b) By the property

Angular Bisector

Angular bisector plane is the plane bisecting the dihedral angle of 2 planes.


How to prove?

Methods:

a) Prove that the angle between (P) and (R) = the angle between (Q) and (R).

b) Prove any point on (R) equidistant from (P), (Q), and the line of intersection.


Axis of a circle

Definition

(d) is the axis of a circle (C) of (d) perpendicular to (C) at O

Property

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

How to prove?

Methods:

a) AB=AC=AD

and

(AO) is perpendicular to (C)

b) By definition

Done by:

Tala Badran

Ghida Ladkani

Marlene Al Abed