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# Intersection of Lines and Planes Help

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## Introduction to the Intersection of Lines and Planes - Angles

### Angles and Distance

Let’s see how the angles between intersecting planes are defined, and how these angles behave. Let’s also see how we can define the angles between an intersecting line and plane, and how these angles behave.

### Angles Between Intersecting Planes

Suppose two planes X and Y intersect in a common line L . Consider line M in plane X and line N in plane Y , such that M L and N L , as shown in Fig. 7-7. The angle between the intersecting planes X and Y is called a dihedral angle , and can be represented in two ways. The first angle, whose measure is denoted by u , is the smaller angle between lines M and N . The second angle, whose measure is denoted by v , is the larger angle between lines M and N .

If only one angle is mentioned, the “angle between two intersecting planes” is usually considered to be the smaller angle u . Therefore, the angle of intersection is larger than zero but less than or equal to a right angle. That is, 0° < u ≤ 90° (0 < uπ /2).

### Adjacent Angles Between Intersecting Planes

Suppose two planes intersect, and their angles of intersection are u and v as defined above. Then if u and v are specified in degrees, u + v = 180°. If u and v are specified in radians, then u + v = π .

Fig. 7-7 . The dihedral angle between the intersecting planes X and Y can be represented by u , the acute angle between lines M and N , or by v , the obtuse angle between lines M and N .

### Perpendicular Planes

Suppose two planes X and Y intersect in a common line L . Consider line M in plane X and line N in plane Y , such that M L and N L , as shown in Fig. 7-7. Then X and Y are said to be perpendicular planes if the angles between lines M and N are right angles, that is, u = v = 90° ( π /2 rad). Actually, it suffices to say that either u = 90° ( π /2 rad) or v = 90° ( π /2 rad).

## Normal Line To A Plane

Let plane X be determined by lines L and M , which intersect at point S . Then line N that passes through plane X at point S is normal (also called perpendicular or orthogonal) to plane X if and only if N L and N M . This is shown in Fig. 7-8. Line N is the only line normal to plane X at point S . Furthermore, line N is perpendicular to any line, line segment, or ray that lies in plane X and runs through point S .

Fig. 7-8 . Line N through plane X at point S is normal to X if and only if N L and N M , where L and M are lines in plane X that intersect at point S .

## Angles Between An Intersecting Line And Plane

Let X be a plane. Suppose a line O , which is not normal to plane X , intersects plane X at some point S as shown in Fig. 7-9. In order to define an angle at which line O intersects plane X , we must construct some objects. Let N be a line normal to plane X , passing through point S . Let Y be the plane determined by the intersecting lines N and O . Let L be the line formed by the intersection of planes X and Y . The angle between line O and plane X can be represented in two ways. The first angle, whose measure is denoted by u , is the smaller angle between lines L and O as determined in plane Y . The second angle, whose measure is denoted by v , is the larger angle between lines L and O as determined in plane Y .

If only one angle is mentioned, the “angle between a line and a plane that intersect” is considered to be the smaller angle u . Therefore, the angle of intersection is larger than zero but less than or equal to a right angle. That is, 0° < u ≤ 90° (0 < uπ /2).

Fig. 7-9 . Angles u and v between a plane X and a line O that passes through X at point S .

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