Properties of the Plane

Excerpts

• Properties of the Plane

  If a line or a plane contains a point, the point is said to be *on* the line or *in* the plane and the line or plane is said to pass *through* the point.

• Properties of the Plane

  While two points determine a straight line it is obvious that two points do not determine a plane.

• Properties of the Plane

  We, therefore, say that three non-collinear points determine a plane, while any number of collinear points fail to determine a plane.

• Properties of the Plane

  In the figure, $PA$ is perpendicular to the plane $M$ and $QA$ is oblique to it.

• Properties of the Plane

  The perpendicular is the shortest distance from a point to a plane.

• Properties of the Plane

  Note that the four vertices of a quadrilateral in space do not necessarily all lie in the same plane.

• Properties of the Plane

  Any line, as $l_1$, in either of two parallel planes, $M$ and $N$, is parallel to the other plane.

• Properties of the Plane

  If two straight lines are cut by three parallel planes, the intercepted segments on one line are proportional to the corresponding segments on the other.

• Properties of the Plane

  Two half-planes meeting in a common edge form a *dihedral angle*.

• Properties of the Plane

  angle may be thought of as *generated* by the rotation of a half-plane about its edge. The magnitude of the angle depends solely upon the *amount of rotation*.

• Properties of the Plane

  point on a plane is the foot of the perpendicular from the point to the plane.

• Properties of the Plane

  The acute angle formed by a straight line with its own projection on a plane is the least angle which it makes with any line in that plane.

• Properties of the Plane

  The trihedral angles $O$ and $O'$ cannot be made to coincide, even though their corresponding parts are equal. This can be illustrated by attempting to put a left glove on the right hand.

• Properties of the Plane

  Show that line-segments included between parallel planes and perpendicular to them are equal, and hence that parallel planes are everywhere equally distant.

Equations

• Properties of the Plane

  \dfrac{AE}{EB} = \dfrac{CG}{GD}

  Two lines cut by three parallel planes are divided in the same ratio: the ratio of the segments on one line equals the ratio of the corresponding segments on the other.

• Properties of the Plane

  EC = ED

  Any point in the bisecting half-plane of a dihedral angle is equally distant from the two faces, measured along the perpendiculars to each face.

• Properties of the Plane

  \angle M-AB-N = \angle M'-A'B'-N'

  Two dihedral angles are equal when their plane angles are equal.

• Properties of the Plane

  \angle CDE = \angle C'D'E'

  Two plane angles are equal when the dihedral angles that they measure are equal.

• Properties of the Plane

  \angle ABD > \angle ABC

  The acute angle a slanting line makes with its own projection on a plane is smaller than its angle with any other line in that plane through the same point.

• Properties of the Plane

  \angle DFE = \angle D'F'E'

  If two trihedral angles have their three face angles equal respectively, the dihedral angles opposite the equal face angles are equal.

Problems

No exercises in this chapter.