# Angles. Projections. Polyhedral angles

*Angles. Angle between: two intersecting straight lines, two parallel*

straight lines, two crossing straight lines. Perpendicular to a plane.

Projections. Projection of a point and a segment to a plane. Dihedral

angle. Linear angle of a dihedral angle. Angle between two planes.

Perpendicular planes. Angle between two perpendicular planes.

Angle between two parallel planes. Polyhedral angle. Plane angles.

Trihedral angle as a minimal polyhedral angle. Parallel sections of

a polyhedral angle.

straight lines, two crossing straight lines. Perpendicular to a plane.

Projections. Projection of a point and a segment to a plane. Dihedral

angle. Linear angle of a dihedral angle. Angle between two planes.

Perpendicular planes. Angle between two perpendicular planes.

Angle between two parallel planes. Polyhedral angle. Plane angles.

Trihedral angle as a minimal polyhedral angle. Parallel sections of

a polyhedral angle.

* Angles. An angle between two intersecting straight lines* is measured as well as in a planimetry ( because it is possible to draw a plane

through these lines ).

*An angle between two parallel straight lines*is accepted equal to 0 or 180°.

*An angle between two crossing straight lines*

AB and CD

*( Fig.70 ) is determined as follows: through any point O the rays OM and ON are drawn so, that OM || AB and*

ON || CD. Then an angle between AB and CD is accepted equal to an angle NOM. In other words, the straight lines AB and

CD are transferred to a new position parallel to themselves until their intersection. Particularly, the point O can be taken in one of the

lines AB or CD, which is immovable in this case.

The straight line AB, intersecting the plane P in the point O ( Fig.71 ), forms different angles ( the angles BOC, BOD, BOE ) with the different straight

lines OC, OD, OE, drawn in the plane P through the point O. If the line AB is perpendicular to the two of these straight lines ( for instance, OC and

OE ), then it is perpendicular to *all* the straight lines, drawn in this plane through the point O. In this case the straight line AB is called

*perpendicular* to the plane P, and the plane P – perpendicular to the straight line AB.

** Projections. **A

*projection of the point*A to the plane P is called a base C of the perpendicular AC, drawn from the point A to the

plane P. A

*projection of*

*the segment*AB to the plane P is the segment CD, ends of which are projections of the points A and B ( Fig.72 ).

It is possible to project not only a straight line, but any curve ABCDE ( Fig. 73 ) to a plane.

Lengths *l* of the projection CD and *a* of the segment AB ( Fig.72 ) are tied by the relation:

** Dihedral angle.** A figure, formed by two half-planes Q and R, going through the same straight line MN ( Fig.74 ), is called a

*dihedral angle*. The straight line MN is called an

*edge*of a dihedral angle; half-planes Q and R – its

*faces*. The plane P,

perpendicular to the edge MN, gives the angle AOB in its intersection with the half-planes Q and R. The angle AOB is called a

*linear*

*angle*of a dihedral angle.

*A linear angle is a measure of its dihedral angle.*

*Angles between planes**. *Two planes are called * perpendicular planes*, if they form a right angle. An angle between two

*parallel planes* is accepted equal to zero. In general case an angle between two planes P and Q ( Fig.75 ) can be measured by an angle,

formed by the straight lines AB and CD, which are perpendicular to the planes P and Q correspondingly.

** Polyhedral angle.** If to draw through the point O ( Fig.76 ) a set of planes AOB, BOC, COD etc., which are consequently intersected

one with another along the straight lines OB, OC, OD etc. ( the last of them EOA intersects the first AOB along the straight line OA ), then we

receive a figure, called a

*polyhedral angle.*The point O is called a

*vertex*of a polyhedral angle. Planes, forming the polyhedral angle

(AOB, BOC, COD, …, EOA), are called its

*faces*; straight lines, along which the consequent faces intersect ( OA, OB, OC, … , OE ) are called

*edges*of a polyhedral angle. Angles AOB, BOC, COD, … , EOA are called its

*plane angles*. The minimal number of faces of a

polyhedral angle is 3 ( the

*trihedral angle*, Fig.77 ).

Parallel planes cut off the proportional segments ( OA:O*a* = OB:O*b* = OC:O*c *= …) on edges of a polyhedral angle ( Fig.78 )

and form the similar polygons ( ABCD and *abcd* ).