EQUALITY OF TRIANGLES. Two SECONDARY CASES Two angles and the side opposite one. 95. THEOREM 19. If two triangles have two angles of one respectively equal to two angles of the other, and the sides opposite one pair of angles equal in each triangle, then the triangles are equal. Let the two triangles ABC and A'B'C' have the angles B, C, and the side AB respectively equal to the angles B', C', and the side A'B'. A A' To prove that the triangles are equal. Place the triangle A'B'C' on ABC so that A'B' falls on its equal AB, and the angle A'B'C' on its equal ABC; then the line B'C' falls in the line BC. Suppose, if possible, that B'C' is not equal to BC, and that the point c' falls on c" instead of falling on C. Then the triangles A'B'C' and ABC" have the sides A'B', B'C', and the included angle A'B'C', respectively equal to AB, BC", ABC". Therefore these triangles are equal; and the angles ACB and A'C'B' are equal (64). But the angles A'C'B' and ACB are equal by hypothesis. Therefore the angles AC"B and ACB are equal; which is impossible (79). Similar reasoning applies if c" falls at the other side of C. Thus the supposition that c' does not fall on C is false; hence c' falls on C, and the triangle A'B'C' on ABC. Therefore the triangles ABC and A'B'C' are equal. 96. Cor. The two perpendiculars drawn from any point in the bisector of an angle to the sides of the angle are equal. Two sides and the angle opposite one. 97. THEOREM 20. If two triangles have two sides of one respectively equal to two sides of the other, and the angle opposite one of these equal to the corresponding angle in the other triangle, then the angles opposite to the other pair of equal sides are equal or supplemental; and if equal, the triangles are equal. Let the two triangles ABC and A'B'C' have the sides AB and BC respectively equal to the sides A'B' and B'C', and the angle BAC equal to the angle B'A'C'. A A A A To FIG. 1 FIG. 2 prove that the angles BCA and B'C'A' are either equal or supplemental. The two sides, AC and A'c', are either equal or unequal. If they are equal (as in fig. 1), the triangles are equal in all their parts (66). If they are unequal (as in fig. 2), let AC be the greater. Lay off AC" equal to 'c'; and draw BC". In the triangle ABC" and A'B'C', the sides AB and AC" and the included angle BAC" are equal respectively to the sides A'B' and 'C' and the included angle B'A'C'. Therefore the angles BC"A and B'C'A' are equal, and the sides BC" and B'C' are equal (64). Therefore BC" equals BC; and hence the angles BCC" and BCC are equal (59). Now BC"C is the supplement of BC"A; therefore BCA, being equal to BC"C, is equal to the supplement of BC''A, and hence equal to the supplement of B'C'A'. 98. Cor. I. If two triangles have two sides of on tively equal to two sides of the other, and the angle one of these sides equal to the corresponding angle in t triangle, then the triangles are equal : (1) If the two angles given equal are right angles o angles; (2) if the angles opposite to the other two equal s both acute, or both obtuse, or if one of them is a right a (3) if the side opposite the given angle in each triang less than the other given side. 99. Cor. 2. The perpendicular from the vertex of celes triangle to the base bisects both the base and the angle. Conversely, the perpendicular bisector of the bas isosceles triangle passes through the vertex. 100. Cor. 3. If the two perpendiculars drawn from to the sides of an angle are equal, then the point is on sector of the angle. (Converse of 96.) EXERCISES 1. Summarize the five cases of the equality of two triangles 2. The bisectors of two adjacent supplemental angles are p dicular to each other. 3. The bisectors of two adjacent conjunct angles are in the straight line. 4. The bisectors of two vertically opposite angles are in the straight line. 5. The lines drawn from the extremities of the base of an iso triangle to the middle points of the opposite sides are equal. 6. The bisector of the vertical angle of an isosceles triangle b the base at right angles. 7. If the bisector of an angle of a triangle is perpendicular t opposite side, the triangle is isosceles. 8. If the perpendicular from a vertex to the opposite side bi that side, then the triangle is isosceles. 9. If two triangles have a common base, and if the vertex of second triangle is within the first, or on a side of the first, ther vertical angle of the first triangle is less than that of the second. SUMMARY OF TYPES OF INFERENCE The foregoing propositions have illustrated various methods of drawing conclusions from given premises. It is now time for us to consider some of the essential features of the modes of inference we have been using, and to see the simple logical principles that underlie them. It will be seen that there are a few general type-forms which appear again and again under various modes of expression; and the student will thus early learn to recognize the logical equivalence of certain statements that differ only in form; also to distinguish between different statements that may seem to be alike; and gradually will come to see the legitimate conclusions that can be inferred from any given premises. 101. Related statements. Hereafter when the word "statement" is used without qualification, it will be understood to mean a simple assertion of the form "A is B." It has been seen that a "theorem" is made up of two such statements placed in a certain relation to each other, the relation of hypothesis (or antecedent) to conclusion (or consequent). When we say that the theorem is "true," we do not mean that either statement is true in itself, but only that the consequent is true whenever the antecedent is true. This may be conveniently expressed by saying that the truth of the consequent is a necessary result of the truth of the antecedent. When any two statements are related to each other so that the truth of each is a necessary consequence of the truth of the other, they are said to be equivalent statements. For instance, the two statements, X is equal to Y, half x is equal to half Y, are equivalent. When either is true, so is the other; and hence when either is false, so is the other. Two statements are said to be partially equivalent when the truth of one is a necessary consequence of the truth of the other, but the truth of the latter not a necessary consequence of the truth of the former. For instance, the two statements, X is greater than Y, X is greater than half Y, are partially equivalent. The truth of the second is a necessary consequence of the truth of the first, but the truth of the first is not a necessary consequence of the truth of the second. When the second is true, the first may be true or false. Two statements are said to be independent when neither is a necessary consequence of the other. For instance, the two statements, X is greater than Y, X is less than double Y, are independent. When either is true, the other may be true or false. Two statements are said to be inconsistent when they cannot both be true at the same time. Inconsistency is of two kinds, opposition and partial opposition. Two statements are said to be opposite, if, when either is true the other is false, and when either is false the other is true. For instance, the two statements, X is equal to Y, X is not equal to Y, are opposite. Two statements are said to be partially opposite, if when either is true the other is false, and when either is false the other may be true or false. For instance, the two statements, X is equal to Y, X is less than Y, |