SECTION I. THE USE OF PLANE GEOMETRY IN SOLID GEOMETRY

In starting solid geometry, all of plane geometry is at the disposal of the student, but great care must be taken that it is used only for those figures for which the plane geometry proofs will hold. For example, it has been proved in plane geometry that Lines parallel to the same line are parallel to each other; but as this was proved only for the case when the three lines were in one plane, it cannot be used for the general case in solid geometry. It cannot be used, for instance, to prove that the back edge of the ceiling of a room is parallel to the front edge of the floor because they are both parallel to the front edge of the ceiling.

The proofs of some propositions in plane geometry would hold even if the figures used did not lie entirely in one plane; as, for example, congruence proofs. In a second kind of proposition, as in theorems concerning triangles, the figures, from the conditions of the proposition, necessarily lie in one plane. Such propositions can be used wherever their figures occur in solid geometry. In a third kind of proposition, where there is nothing in the conditions to show whether or not the figure is in one plane, it must be proved to lie in one plane before the plane geometry proof can be assumed to hold. An example is the proposition Lines perpendicular to the same line are parallel. By looking at the edges meeting at the corner of a room it will be seen that this proposition does not hold for the general case.

The propositions of plane geometry that are of most value in solid geometry will be summarized under the three heads indicated in the last paragraph.

A. SOME PARTS OF PLANE GEOMETRY THAT CAN BE USED IN SOLID GEOMETRY WHETHER OR NOT THE ENTIRE FIGURE IS IN ONE PLANE

I. (a) All the definitions and axioms.

(b) All right angles are equal; all straight angles are equal.

(c) Complements, supplements, or explements of equal angles are equal.

II. FIGURES CONGRUENT.

(a) The definition: two figures are congruent if they can be made to coincide.

(b) Two triangles are congruent if they have three parts of one, of which at least one is a side, equal to the corresponding parts of the other; unless those three parts are two sides and an acute angle opposite one of them.

III. SIMILAR FIGURES.

(a) Two polygons are similar (1) if their corresponding sides are proportional and their cor

responding angles are equal; (2) if they are similar to the same polygon.

(b) Two triangles are similar if they have (1) two angles equal; (2) two sides of one proportional to two sides of the other, and the included angle equal; (3) their sides correspondingly proportional.

(c) Areas of similar figures are proportional to the squares of any two corresponding sects.

IV. OTHER COMPARISONS.

(a) If two triangles have two sides of one equal to two sides of the other, then the third side of one is correspondingly greater than, equal to, or less than, the third side of the other, ac

cording as its opposite angle is greater than, equal to, or less than, the opposite angle of the other, and conversely.

(b) Areas of triangles having an angle of one equal to an angle of the other are to each other as the products of the including sides.

B. PLANE GEOMETRY PROPOSITIONS THAT CAN BE USED IN SOLID GEOMETRY BECAUSE THE NATURE OF THE FIGURE MAKES IT LIE ENTIRELY IN ONE PLANE

See § 9. Figures that lie entirely in one plane are a triangle, a line and a point outside, two intersecting lines, two parallel lines and therefore a parallelogram, and a circle. I. PROPOSITIONS ABOUT A TRIANGLE.

(a) If one side of a triangle is greater than, equal to, or less than, a second side, the angle opposite the first side is respectively greater than, equal to, or less than, the angle opposite the second side, and conversely.

(b) The sum of two sides of a triangle is greater than the third side; their difference is less than that side.

(c) The perpendicular bisectors of the sides of a triangle meet in a point (called the circumcenter) equidistant from the vertices.

(d) The bisectors of the interior angles of a triangle meet in a point (called the incenter) equidistant from the sides.

(e) The medians of a triangle meet in a trisection point of each. In an equilateral triangle,

the medians are also the altitudes.

(ƒ) A line parallel to one side of a triangle cuts the other sides proportionally, and con

versely; it is in the same ratio to the side to

which it is parallel as the sects it cuts off on

the other sides from their common vertex are to those sides.

(g) In a right triangle with the altitude drawn to the hypotenuse, (1) the triangles are similar; (2) the altitude is the mean proportional between the sects of the hypotenuse; (3) each leg is the mean proportional between the hypotenuse and its own projection upon the hypotenuse.

II. PROPOSITIONS ABOUT PARALLELS AND PARAL

LELOGRAMS.

(a) Definition: lines in the same plane that never meet are parallel.

(b) Through a point there can be but one parallel
to a given line.

(c) If two parallels are cut by a transversal, two
sets, each of four equal angles, are formed,
the angles of one set being supplemental to
the angles of the other set, and conversely.
(d) In a parallelogram (1) the opposite sides are
equal, (2) the opposite angles are equal,
(3) either diagonal divides it into congru-
ent triangles, (4) the diagonals bisect each
other.

(e) A quadrilateral is a parallelogram if (1) the
opposite sides are parallel, (2) two opposite

sides are equal and parallel, (3) the diago-
nals bisect each other.

(f) Parallelograms having equal bases, and lying
between the same parallels, are equivalent.
(g) The sect between the midpoints of the legs of
a trapezoid is one half the sum of the bases.

III. PROPOSITIONS IN REGARD TO A LINE AND AN

EXTERNAL POINT.

(a) There can be but one perpendicular from an external point to a line.

(b) From an external point to a line, (1) the perpendicular is the shortest line; (2) obliques that make equal angles with the perpendicular or with the given line, or that cut off equal distances from the foot of the perpendicular, are equal, and conversely; (3) obliques that make unequal angles with the perpendicular, or with the given line, or that cut off unequal distances from the foot of the perpendicular, are unequal, the one that makes the greater angle with the perpendicular, or that cuts off the greater distance, being longer, and conversely.

IV. PROPOSITIONS ABOUT CIRCLES.

(a) Radii, or diameters, of a circle are equal.
(b) The diameter is the greatest chord; it bisects
the circle and its circumference. If it is
perpendicular to a chord, it bisects the
chord, and conversely.

(c) One, and but one, circle can be inscribed in,
or circumscribed about a given triangle.
(d) A central angle is measured by its arc; an
inscribed angle, by one half its arc.

(e) Equal chords subtend equal arcs, and are equi-
distant from the center, and conversely.

The greater chord subtends the greater arc, and is nearer the center, and conversely. (f) A tangent is perpendicular to the radius drawn to the point of contact.