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SOLID GEOMETRY

BOOK WI OF GEOMETRY
LINES AND PLANES IN SPACE
I. REFERENCES TO PLANE GEOMETRY

1. Nature of Solid Geometry. In plane geometry we consider figures which lie in a plane, study their properties and relations, and measure their dimensions and areas. Such figures are, in general, two-dimensional.

In solid geometry we shall consider not only figures of one dimension and two dimensions, but also threedimensional figures, such as cubes and spheres.

In considering the theory of the subject we shall not need to construct the solid figures by means of the straightedge and compasses, and hence we shall not discuss any problems of construction.

2. References to Plane Geometry. In proving the propositions and exercises of solid geometry, we shall often need to refer to the definitions, assumptions, and propositions of plane geometry as authorities for the statements which form the various steps in the proofs. Pages 2–18 contain such references, grouped in a convenient manner, as are necessary for this purpose.

The definitions of such familiar terms as vertex, side, altitude,

radius, diameter, and so on are omitted.

3. Preliminary Definitions. 1. An angle is a figure formed by two rays which proceed from the same point.

The term ray is applied to a straight line which begins at a certain point and extends indefinitely.

2. A straight angle is an angle whose arms extend in opposite directions so as to be in one straight line.

The sum of the angles about a point is two straight angles, or 360°.

3. A right angle is half of a straight angle.

An acute angle is less than a right angle; an obtuse angle is greater.

4. Either of two lines is perpendicular to the other if one meets the other so as to make a right angle with it.

Remember that any definition may be inverted.

5. Equal angles are angles which can be placed one on the other so that they coincide.

6. Two angles are complementary if their sum is a right angle.

7. Two angles are supplementary if their sum is a straight angle.

8. Vertical angles are two angles which have the same

vertex and the sides of one in prolongation of the sides of the other.

9. A plane is a surface such that a straight line joining any two of its points lies wholly in the surface.

10. A rectilinear figure is a figure which lies wholly in one plane and which represents a surface bounded by segments of straight lines.

11. A triangle is a rectilinear figure of three sides.

The words isosceles, equilateral, equiangular, right, acute, and obtuse, as applied to triangles, are already familiar to the student.

§§ 3–5 DEFINITIONS AND AXIOMS 3

4. Axioms. 1. If equals are added to equals, the sums are equal. An aziom is a general statement admitted without proof.

2. If equals are subtracted from equals, the remainders are equal.

3. If equals are multiplied by equals, the products are equal.

4. If equals are divided by equals, the quotients are equal.

The divisor must never be zero, division by zero having no meaning.

5. A number or magnitude may be substituted for its equal.

6. Like powers or like roots of equal numbers are equal.

7. If equals are added to or subtracted from unequals, or if unequals are multiplied or divided by equals, the results are unequal in the same order.

8. If unequals are added to unequals in the same order, the sums are unequal in the same order; if unequals are subtracted from equals, the remainders are unequal in reverse order.

9. If the first of three quantities is greater than the second, and the second is greater than the third, then the first is greater than the third.

10. The whole is greater than any of its parts and is equal to the sum of all its parts.

5. Postulates. 1. One and only one straight line can be drawn through two distinct points. A postulate is a geometric statement admitted without proof.

Post. 1 may also be stated thus: Two distinct points determine a line, or, Two straight lines cannot intersect in more than one point.

2. A straight-line segment can be produced to any required length.

3. A straight-line segment is the shortest path between two points.

4. Any figure can be moved without altering its shape or size.

5. All straight angles are equal and all right angles are equal.

6. Angles which have equal complements or equal Supplements are equal.

7. There is one and only one line which, passing through a given point, is perpendicular to a given line.

6. Wertical Angles. If two lines intersect, the vertical angles are equal.

7. Congruent Triangles. 1. Congruent figures are figures which can be made to coincide in all their parts.

2. In two congruent figures the parts of one figure are equal respectively to the corresponding parts of the other.

3. If two sides and the included angle of one triangle are equal respectively to two sides and the included angle of another, the triangles are congruent.

4. If two angles and the included side of one triangle are equal respectively to two angles and the included side of another, the triangles are congruent.

5. If the three sides of one triangle are equal respectively to the three sides of another, the triangles are congruent.

6. If the hypotenuse and an adjacent angle of one right

triangle are equal respectively to the hypotenuse and an adjacent angle of another, the triangles are congruent.

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7. If the hypotenuse and a side of one right triangle are equal respectively to the hypotenuse and a side of another, the triangles are congruent.

8. If two angles and a side of one triangle are equal respectively to two angles and the corresponding side of another, the triangles are congruent.

8. Angles of a Triangle. 1. In an isosceles triangle the angles opposite the equal sides are equal. 2. If a triangle has two equal angles, the sides opposite these angles are equal. 3. An equilateral triangle is equiangular. 4. The sum of the three angles of a triangle is a straight angle. 5. An exterior angle of a triangle is greater than either nonadjacent interior angle. An eacterior angle of a figure is the angle included between one side

and an adjacent side produced.

6. An exterior angle of a triangle is equal to the sum of the two nonadjacent interior angles.

9. Parallel Lines. 1. Parallel limes are lines which lie in the same plane and cannot meet, however far they may be produced.

2. Through a given point only one line can be drawn parallel to a given line. 3. A transversal is a line cutting two or more others.

When a transversal cuts two lines, it forms eacterior angles, interior angles, alternate angles, and corresponding angles. The student is familiar with the usual figure of a transversal cutting parallels.

4. If two parallel lines are cut by a transversal, the alternate angles are equal.

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