back to Solid Geometry

More on plane, line and angles

Lines not on the same plane

If two lines are neither parallel nor intersect, then these two line are called that lines are not in the same plane.

In the figure about, line n is on the plane S but line m is not on the plane S.

In the figure above, line m is on the plane T and the line n is on the plane S. Plane S and plane T intersect at the line l.

Three lines parallel axiom

If two lines parallel to the third line, then the two lines are parallel each other.

If line a parallel to line b and line c parallel to line b, then line a parallel to line c.

Note: line a , line b and line c are all on the same plane S.

If line d parallel to line e and line f parallel to line e, then line d parallel to line f.

Note: line d and line e are on the plane S. Line f is on the plane T. Line d and line f are on the different plane.

Equal angles theorem

If two sides of an angle parallel to and have the same direction with two sides of another angle, then the two angles are equal.

In the figure above, AB parallel DE, AC parallel to DF, then angle BAC parallel to angle EDF.

Solid quadrilaterals

The quadrilateral whose four vertex are not on the same plane is called the solid quadrilateral.

In the figure above, Points A, B, and C are on the same plane. D is a point outside the plane ABC. Connect DA, DB and DC, which make a solid quadrilateral ABCD.

Angle formed by two lines that are not in the same plane

Definition: line a and line b are two lines that are not in the same plane, pass through any point O in the solid, make line c parallel to lines a and d parallel to line b, then the angle formed by lines c and d is the angle formed by two lines a and b that are not on the same plane.

For example, E is a point on line segment DC, F is a point on line segment AB, connect EF. What is the angle formed by lines DB and EF? (Note: line segment DB and EF are not in the same plane).

To solve this problem, drawing a line that pass the point F and parallel to DB which intersects AD at point G, then the angle GFE is the angle formed by DB and EF.

Two lines that are not on the same plane perpendicular to each other

Line a lies on a plane and line b lies on another plane. Line a perpendicular to line b.

Common perpendicular line segment of two lines that are not in the same plane

Line a lies on a plane and line b lies on another plane. Line c perpendicular and intersects to line a. Line c perpendicular and intersects to line b. So line c is called the common perpendicular line segment of two lines that are not in the same plane.

The distance between two lines that are not on the same plane

Line a lies on a plane and line b lies on another plane. Line c is perpendicular to line a. Line c is also perpendicular to line b. The line segment c below line a and above line b is called the distance of two lines that are not on the same plane.

Solid lines and planes

Line parallel to plane

If a line has no common point with a plane, then the line parallel to the plane.

In the figure above, line a has no common point with the plane S, so the line a parallel to the plane S.

The position relation of the line and plane

1. Lines on the plane

If a line lies on a plane, then both the line and the plane has infinite common points.

When a line lies on a plane, both the line and the plane has infinite common points.

2. line intersects with a plane

If a line intersects a plane, then both the line and the plane has and only has a common point.

In the figure above, line a intersects the plane S at the point A. That is, A is the common point in the line and the plane.

3. Line parallel to plane

When a line parallel to a plane, both the line and the plane has no common point.

Determine whether a line parallel to a plane theorem

If a line outside a plane parallel to a line on the plane, then the line parallel to the plane.

In the figure above, line a lies outside the plane S. Line b lies on the plane S. If line a parallel to the line b, then line a parallel to the plane S.

Property of line parallel to plane theorem

If a line parallel to a plane, another plane that pass this line intersects the plane, then the line parallel to the plane.

In the figure above, if line a parallel to plane Q and the plane S that contains the line a pass through the plane Q, then the line a parallel to the plane Q.

Line perpendicular to a plane

If a line perpendicular to any lines on plane, then this line perpendicular to the plane.

In the figure above, if line b perpendicular to any lines on the plane S, then the line b perpendicular the plane S.

Determine whether a line perpendicular to a plane theorem 1

If a line outside a plane perpendicular to two intersected line on a plane, then this line perpendicular to the plane.

In the figure above, e is a line outside the plane T, c and d are two lines on the plane T. If the line e perpendicular to the lines c and d, then the line e perpendicular to the plane T.

Determine whether a line perpendicular to a plane theorem 2

If one of two parallel lines perpendicular to a plane, the other line is also perpendicular to the plane.

In the figure above, line b parallel to line c. If the line b perpendicular to the plane S, then the line c also perpendicular to the plane S.

Property of lines perpendicular to plane theorem

If two lines parallel to a plane, then these two lines are parallel to each other.

In the figure above, if line e perpendicular to the plane S and the line f also perpendicular to the plane S, then line e parallel to line f.

The distance from a point to a plane

In the figure below, drawing a line pass through the point A and perpendicular to the plane T, which intersects the plane T at the point B, the length of AB is the distance from the point A to the plane T.

The distance from a line to a plane

In the figure below, line b parallel to the plane S. C and D are points on line b. Draw a line that pass through the point C and perpendicular to the plane S, which intersects the plane S at the point D, the distance CD is the distance from line b to the plane S. Draw a line that pass through the point E and perpendicular to the plane S, which intersects the plane S at the point F, the distance EF is the distance from line b to the plane S. The length of CD equals the length of EF.

The projection of a point to a plane

In the figure below, C is a point outside the plane S. D is the projection of the point C on the plane S. The length of CD is called the perpendicular segment from the point C to the plane S.

Inclined line to a plane

In the figure below, line b intersects but not perpendicular to the plane T. The line b is called the inclined line to the plane T. The point A is called the inclined point. B is a point on the line b, the distance AB is called the inclined segment.

Projection of the inclined line to a plane

In the figure below, B is a point on the inclined line b, drawing a line pass through the point B and perpendicular to the plane T, which intersects the plane T at point C. The point C is called the perpendicular point. AB is the inclined line segment. A is called the inclined point. Connect AC, AC is the projection of the inclined line segment AB.

Relation among the perpendicular segment, inclined line segment and projection

1. If two projections are equal, then their inclined lines are equal.

2. If two inclined lines are equal, then their projections are equal.

3. Perpendicular segment is the shortest segment among all inclined segments.

Angle formed by line and plane

In the figure below, b is an inclined line and c is the projection of the inclined lined b to the plane T. The angle 1 formed by the inclined line and the projection is called the angle formed by a line and a plane.

1. When a line perpendicular to a plane, the angle formed by the line and the plane is a right angle.

2. When a line parallel to a plane or the line lies on the plane, then the angle formed by the line and the plane is 0^o.

The angle a formed by a line and a plane is in the range of 0^o <= a <= 90^o.

The smallest angle theorem

In the figure below, the angle 1 formed by the projection c and the inclined line b. The angle 2 formed by the inclined line and any line which lies on the plane T and pass the inclined point. angle 1 < angle 2. The angle formed by the projection and the inclined line is the smallest angle.

Three perpendicular lines theorem

In the figure below, b is the inclined line to the plane T and c is the projection of the inclined line b to the plane T. There is line d on the plane T. If the line d perpendicular to the projection c, then the line d is also perpendicular the inclined line b.

Three perpendicular lines inverse theorem

In the figure below, d is a line on the plane T. b is the inclined line to the plane T. c is the projection of the inclined line b to the plane T. If d perpendicular to the inclined line b, then d is also perpendicular to the projection c.