Lines and Planes in Space
LINES AND PLANES IN SPACE
PLANE – is a surface such that if any two points on it are joined by a straight line, the line lies wholly on the surface.
POSTULATE 1. Through three points not on the same straight line, one and only one plane can be passed.
A plane is determined by any of the following conditions:
1. Three non-collinear points;
2. A line and a point not on the line;
3. Two intersecting lines; or
4. Two parallel lines.
POSTULATE 2. If two planes meet, they have at least two distinct points in common.
PARALLEL LINES- lines that lie on the same plane and cannot meet however far they are produced. PARALLEL PLANES – planes which do not intersect however far they are produced. FOOT OF A LINE - the point of intersection of a line and a plane or of a line and a second line. NORMAL TO A PLANE – a line perpendicular to a plane.
The projection of a point on the plane is the foot of the perpendicular let fall from the point to the plane. The projection of a line on the plane is the locus of the projection of all its points.
THEOREMS:
1. If two planes intersect, their intersection is a line.
2. Through one straight line an infinite number of planes may be passed.
3. The intersections of two parallel planes by a third plane are parallel lines.
4. If two straight lines are parallel, a plane containing one and only one line is parallel to the other line.
5. If a straight line is parallel to a plane, and another plane containing this line intersects the given plane, the intersection is parallel to the given line.
6. If parallel planes intersect two lines, the corresponding intercepts are proportional.
7. Two lines parallel to the same line are parallel to each other.
8. If two angles not in the same plane have their corresponding sides parallel to each other and extending in the same direction from their vertices, the angles are equal and the planes are
PLANE – is a surface such that if any two points on it are joined by a straight line, the line lies wholly on the surface.
POSTULATE 1. Through three points not on the same straight line, one and only one plane can be passed.
A plane is determined by any of the following conditions:
1. Three non-collinear points;
2. A line and a point not on the line;
3. Two intersecting lines; or
4. Two parallel lines.
POSTULATE 2. If two planes meet, they have at least two distinct points in common.
PARALLEL LINES- lines that lie on the same plane and cannot meet however far they are produced. PARALLEL PLANES – planes which do not intersect however far they are produced. FOOT OF A LINE - the point of intersection of a line and a plane or of a line and a second line. NORMAL TO A PLANE – a line perpendicular to a plane.
The projection of a point on the plane is the foot of the perpendicular let fall from the point to the plane. The projection of a line on the plane is the locus of the projection of all its points.
THEOREMS:
1. If two planes intersect, their intersection is a line.
2. Through one straight line an infinite number of planes may be passed.
3. The intersections of two parallel planes by a third plane are parallel lines.
4. If two straight lines are parallel, a plane containing one and only one line is parallel to the other line.
5. If a straight line is parallel to a plane, and another plane containing this line intersects the given plane, the intersection is parallel to the given line.
6. If parallel planes intersect two lines, the corresponding intercepts are proportional.
7. Two lines parallel to the same line are parallel to each other.
8. If two angles not in the same plane have their corresponding sides parallel to each other and extending in the same direction from their vertices, the angles are equal and the planes are