11. Boundaries. The boundaries (or boundary) of a solid are surfaces. The boundaries (or boundary) of a surface are lines. The boundaries of a line are points. These boundaries can be no part of the things they limit. A surface is no part of a solid; a line is no part of a surface; a point is no part of a line. 12. Motion. If a point moves, its path is a line. Hence, if a point moves, it generates (describes or traces) a line; if a line moves (except upon itself), it generates a surface; if a surface moves (except upon itself), it generates a solid. NOTE. Unless otherwise specified the word "line" hereafter means straight line. 13. A plane angle is the amount of divergence of two straight lines that meet. The lines are called the sides of the angle. The vertex of an angle is the point at which the lines meet. 14. Adjacent angles are two angles that have the same. vertex and a common side between them. 15. Vertical angles are two angles that have the same vertex, the sides of one being prolongations of the sides of the other. 16. If one straight line meets another and makes the adjacent angles equal, the angles are right angles. 17. One line is perpendicular to another if they meet at right angles. Either line is perpendicular to the other. The point at which the lines meet is the foot of the perpendicular. Oblique lines are lines that meet but are not perpendicular. 18. A straight angle is an angle whose sides lie in the same straight line, but extend in opposite directions from the vertex. OBTUSE ANGLE ACUTE COMPLEMENTARY SUPPLEMENTARY ANGLES 19. An obtuse angle is an angle that is greater than a right angle. An acute angle is an angle that is less than a right angle. An oblique angle is any angle that is not a right angle. 20. Two angles are complementary if their sum is equal to one right angle. Two angles are supplementary if their sum is equal to two right angles. Thus the complement of an angle is the difference between one right angle and the given angle. The supplement of an angle is the difference between two right angles and the given angle. 21. A degree is one ninetieth of a right angle. The degree is the familiar unit used in measuring angles. It is evident that there are 90° in a right angle; 180° in two right angles, or a straight angle; 360° in four right angles. 22. Notation. A point is usually denoted by a capital letter, placed near it. A line is denoted by two capital letters, placed one at each end, or one at each of two of its points. Its length is sometimes represented advantageously by a small letter written near it. Thus, the line AB; the line RS; the line m. A R S B m An angle is usually denoted by three capital letters, placed one at the vertex and one on each side. If only one angle is at a vertex, the capital letter at the vertex is sufficient to designate the angle. Sometimes it is advantageous to name an angle by a small letter placed within the angle. The word "angle" is usually denoted by the symbol “Z” in geometrical processes. 66 It is important that in naming an angle by the use of three letters, the vertex-letter should be placed between the others. The size of an angle does not depend upon the length of the sides, but only on the amount of their divergence. Thus, ≤ x = ≤ P and ≤ P is the same as ▲ APR or ZAPS or ZBPS, etc. An angle is said to be included by its sides. An angle is bisected by a line drawn through the vertex and dividing the angle into two equal angles. TRIANGLES 23. A triangle is a portion of a plane bounded by three straight lines. These lines are the sides. The vertices of a triangle are the three points at which the sides intersect. The angles of a triangle are the three angles at the three vertices. Each side of a triangle has two angles adjoining it. The symbol for triangle is “▲”. 66 The base of a triangle is the side on which the figure appears to stand. The vertex of a triangle is the vertex opposite the base. The vertex angle is the angle opposite the base. 24. Kinds of triangles: A scalene triangle is a triangle no two sides of which are equal. 25. The hypotenuse of a right triangle is the side opposite the right angle. The sides forming the right angle are called the legs. In an isosceles triangle the equal sides are sometimes called the legs, and the other side, the base. 26. Homologous Parts. If two triangles have the three angles of one equal respectively to the three angles of the other, the pairs of equal angles are homologous. Homologous sides in two triangles are opposite the homologous angles. 27. Homologous parts of equal figures are equal. If the triangles DEF and HIJ are equal in all respects, ZD is homologous to, and =▲ H, hence EF is homologous to, and = IJ. And ZE is homologous to, and I, hence, DF is homologous to, and = HJ, and so on. SUPERPOSITION. SYMBOLS 28. Equality and coincidence. Two geometrical figures are equal if they can be made to coincide in all respects. Angles coincide, and are equal, if their vertices are the same point and the sides of one angle are identical with the sides of the other. Superposition is the process of placing one figure upon another. This method of showing the equality of two geometrical figures is employed only in establishing fundamental principles. 29. Symbols. The usual symbols and abbreviations employed in geometry are the following: AXIOM, POSTULATE, AND THEOREM 30. An axiom is a truth assumed to be self-evident. It is a truth which is received and assented to immediately. 31. AXIOMS. 1. Magnitudes that are equal to the same thing, or to equals, are equal to each other. 2. If equals are added to, or subtracted from, equals, the results are equal. 3. If equals are multiplied by, or divided by, equals, the results are equal. [Doubles of equals are equal; halves of equals are equal.] 4. The whole is equal to the sum of all of its parts. 5. The whole is greater than any of its parts. 6. A magnitude may be displaced by its equal in any process. [Briefly called "substitution."] 7. If equals are added to, or subtracted from, unequals, the results are unequal in the same sense. 8. If unequals are added to unequals in the same sense, the results are unequal in that sense. 9. If unequals are subtracted from equals, the results are unequal in the opposite sense. |