485. The projection of a point upon a plane is the foot of the perpendicular from the point to the plane.

The projection of a line upon a plane is the line formed by the projections of all the points of the given line.

486. A plane is determined if its position is fixed and if that position can be occupied by only one plane.

PRELIMINARY THEOREMS

487. THEOREM. If two points of a straight line are in a plane, the whole line is in the plane. [Def. 481.]

488. THEOREM. A straight line can intersect a plane in only one point. [See 487.]

489. THEOREM.

If a line is perpendicular to a plane, it is perpendicular to every line in the plane drawn through its foot. [See 483.]

490. THEOREM. Through one straight line any number of planes may be passed.

Because, if we consider a plane containing a line AB to revolve about AB, it may occupy an indefinitely great number of positions. Each of these will be a different plane containing AB.

A

-B

491. THEOREM. Through a fixed straight line and an external point a plane can be passed.

Because, if we pass a plane containing this line AB, it may be revolved about AB until it contains the given point.

492. THEOREM.

A straight line and an external point determine

a plane. [See 491, 486.]

493. THEOREM. plane.

Three points not in a straight line, determine a

Because two of the points may be joined by a line; then this line and the third point determine a plane. [See 492.]

494. THEOREM. Two parallel lines determine a plane. [See 91 and 492.]

495. THEOREM.

Two intersecting straight lines determine a plane.

Because one of these lines and a point in the second line determine a plane (492).

And this plane contains the second line (487).

496. THEOREM.

If two planes are parallel, no line in the one can

meet any line in the other. [Def. 484.]

In

NOTE. A plane is represented to the eye by a quadrilateral. some positions it appears to be a parallelogram, and in others, a trapezoid. The eye, however, must be aided by the imagination in really understanding the diagrams of Solid Geometry. Thus, in the adjoining figure, the line CN is perpendicular to the plane FR, and it is perpendicular to every line in FR drawn through N. Consider several lines drawn through a point on the floor, and a cane, CN, occupying a vertical position, so that it is perpendicular to all of these lines. Then every

F

X

-X

-X

X

angle CNX is a right angle, though to the unskilled eye they do not all appear to be right angles in the diagram. The object of all geometrical diagrams is that the eye may assist the mind in grasping truths or in developing logical demonstrations, and the student should thoroughly examine every figure until he completely understands the relative positions of its parts, and thus trains his eye to see three dimensions represented in a plane. Photography accomplishes this, and we should be as familiar with the significance of a geometrical diagram, as with a picture.

When, during the process of a demonstration or elsewhere, it becomes necessary to employ a plane not already indicated, it is customary to pass such a plane, or to conceive it constructed.

497. THEOREM. If two planes intersect, their intersection is a

MN (?). That is, AB is common to both planes.

B

N

S

Now, if there were a point outside of AB, in both planes, these planes would coincide (?) (492).

That is, AB contains all points common to planes MN and RS.

Hence, AB is the intersection (482).

That is, the intersection is a straight line.

Q.E.D.

If two straight lines are parallel, a plane contain

498. THEOREM.

ing one, and only one, is parallel to the other line.

Given: lines AB and CD; plane MN containing CD.

To Prove: plane MN || to line AB.

Proof: AB and CD are in the same plane AD (?) (91). Plane AD intersects plane MN in CD (?) (497).

M

B

N

If AB ever meets MN, it must meet MN in CD; but AB can never meet CD (Hyp.).

.. AB can never meet MN, and AB is || to MN (?) (484).

Q.E.D.

499. THEOREM. 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. A

Given: AB to MN; plane AD containing AB and intersecting plane MN in CD.

To Prove: AB || to CD.

Proof: AB and CD are in the same plane AD (Hyp.). If AB ever meets CD, it must

M

B

N

meet CD in plane MN; but AB can never meet MN (Hyp.). .. AB can never meet CD, and AB is to CD (?) (91).

Ex. 1. Why are the usual folds in a sheet of paper straight lines? Ex. 2. If a rod is held parallel to the pavement, why is the shadow parallel to the rod ?

501. THEOREM. A straight line perpendicular to each of two straight lines at their intersection is perpendicular to the plane of the lines.

Given: AF to BF and CF

at F; plane MN containing BF and CF.

To Prove: AF1 to plane MN. Proof: In plane MN draw BC; draw also DF from F to any point, D, in BC.

Prolong AF to X, making FX = AF, and draw AB, AD, AC, BX, DX, CX.

M

N

Now, BF and CF are 1 to AX at its midpoint (Hyp. and Const.).

In ABC and BCX, AB = BX, AC = CX (?) (67), and BC= BC (?). ..▲ ABC=▲ BCX (?) (58).

Also, in ▲ ABD and BDX, ≤ ABC= ≤ CBX (?) (27), BD=BD (?), and AB = BX (?).

.. ▲ ABD=▲ BDX (?) (52) and AD= DX (?).

.. DF is to AX (?) (70).

That is, AF is

to all lines in MN through F.

Consequently, AF is to plane MN (?) (483).

Q.E.D.

502. THEOREM. All straight lines perpendicular to a line at one

point are in one plane, which is perpendicular to this line at this point.

Given: AB1 to BC, BD, BE, etc.; plane MN containing BC and BD.

To Prove: BE is in the plane MN and MN is 1 to AB at B.

Proof Pass plane AE containing AB and BE, and intersecting MN in line BX.

Now, AB is to MN (?) (501).

M

C

A

X

B

E