TRIANGLES 77. Parts of a Triangle. A triangle has six parts, three sides and three angles. An angle and a side are spoken of as opposite to each other when the side is not one of the arms of the angle. A side is sometimes spoken of as included between two angles, and an angle as included between two sides, when the order in which they lie is meant. 78. Vertex Angle. The angle opposite the base of a triangle is called the vertex angle, or the vertical angle. 79. Equality of Sides. If a triangle has two equal sides, it is called isosceles; if no equal sides, scalene. In an isosceles triangle, the equal sides are sometimes called legs, the third side the base. 80. Angles of a Triangle. If all the angles of a triangle are acute, it is called an acute-angled triangle; if one angle is right, it is called a right triangle; and if one angle is obtuse, it is called an obtuse-angled triangle. In a right triangle the side opposite the right angle is called the hypotenuse, and the other sides the legs. 81. Lines of a Triangle. There are four kinds of lines of importance in work with a triangle: the bisectors of the angles, the perpendicular bisectors of the sides, the altitudes, and the medians. The first two explain themselves; an altitude is a perpendicular from a vertex to the opposite side, and a median is a line from a vertex to the midpoint of the opposite side. If the altitude of a triangle is spoken of, the altitude to the base is meant. SECTION VI. INEQUALITIES 82. Axiom of Unequals. The whole is greater than any of its parts. 83. Inequalities. There are certain truths relating to statements of inequality that depend very closely on the general axioms. Their proofs are given under the head of Inequalities in nearly all Algebras, so they will not be considered here. They are used for those magnitudes for which equality axioms 2-5 are used. (1) If equals are added to, taken from, multiplied by, or divided into, unequals, the results are unequal in the same sense. (That is, the greater quantity remains greater after the operation is performed.) (2) If unequals are taken from, or divided into equals, the results are equal in the opposite sense. (3) If unequals are added to, or multiplied by, unequals in the same sense, the results are unequal in the same sense. (4) If the first of several magnitudes is greater than the second, the second greater than the third, the third greater than the fourth, and so on, then the first is greater than the last. NOTE. (4) holds for "the first less than the second," etc.; also any pairs might be equal without changing the result, if there is at least one inequality. These statements are all in regard to positive magnitudes; they are not all true when negative quantities are used. WARNING. Unequals should not be taken from, or divided into, unequals, for the results cannot, in general, be determined. 84. Propositions. Proposition is a general term including: (1) Theorem, which is a truth to be proved. (2) Corollary, which is also a truth to be proved, but generally one that follows quite directly, and often very simply, from a known truth - most often from a theorem that has just been proved. (3) Problem, or Construction Theorem, which requires that a certain figure be drawn from given parts. A more definite understanding of problem will be given in § 177. 85. Parts of a Theorem. A theorem is composed of the condition, or hypothesis, and the conclusion. It usually takes the form, If a certain condition is true, a certain conclusion is also true. (See § 2.) 86. Proof of a Proposition. To prove any proposition, the student has certain materials from which to work, namely the foregoing definitions, axioms, and truths concerning them; the theorems preceding the one that is being proved; and the condition of the proposition in question. From these known truths the proof must be deduced, and the required conclusion must be reached. The proof must always be a logical one; the truth of a proposition must not be judged by measurements — as in Concrete Geometry or from the appearance of the figure, for such methods have no place in this subject. It might be said, however, that a carefully drawn figure will some times give the idea which suggests the correct proof, although the appearance of the figure cannot be quoted as authority for the truth of any statement. 87. Order of Proof. The following order of proof has been found very convenient, and the student is advised to follow it in all work. Given. To prove. figure. STATEMENT OF THEOREM Figure Condition, in terms of the letters of the figure. Proof. I. The proof, in numbered steps, with the authority for each step following it. NOTES. Any special case or interesting fact concerning the theorem. COROLLARIES. Those connected with the theorem. Examples of this form will be found in the proofs given later. 88. Classification of Theorems. All theorems of Plane Geometry may be divided into certain groups which might be called classes; for example, some theorems prove angles equal, others prove lines unequal, still others prove triangles congruent, etc. It is evident that any theorem of a certain class must depend upon something which will prove the particular result desired; that is, upon something in its own class, unless it can be obtained by logic from another class. Except in the cases of logic, which are readily recognized, each geometric truth depends directly upon some preceding truth of its own class, and the foundation truths of each class are the definitions and axioms that concern the things used. 89. Important Classes Already Started. (1) Congruence. To prove figures congruent, they must be shown to coincide; the axiom of motion can be used to suppose one of two figures placed on the other in order to test the coincidence. (2) Angles Equal. Right angles, straight angles, perigons, complements of equal angles, supplements of equal angles, explements of equal angles, vertical angles. (3) Magnitudes Unequal. The inequality axiom: after having obtained one inequality, the inequality statements can be used. It follows from this that the only way to prove two things unequal at this stage of the work is to show that one is a part of the other, or that one equals a part of the other. (4) Line Straight. A line is straight if its sects are the arms of a straight angle; the bisectors of vertical angles lie in a straight line. (5) Lines Perpendicular. If the adjacent angles formed by them are equal; if they bisect adjacent supplemental angles. If they go (6) Lines Coincide (or are determined). through the same two points, if they bisect the same angle, if they are perpendicular to the same line at the same point (in a plane). (7) Points Coincide (or are determined). If they are intersections of the same two lines, if they are midpoints of the same sect, if they are correspondingly placed points of equal sects which are made to coincide. (8) Lines Equal. This would be a case of congruence; the lines would have to be shown to coincide. These are, of course, not all the classes to be found in Plane Geometry, but they serve to show that a founda |