75. A corollary is a truth easily deduced from the proposition to which it is attached. 76. A scholium is a remark upon some particular feature of a proposition. 77. The converse of a theorem is formed by interchanging its hypothesis and conclusion. Thus, If A is equal to B, C is equal to D. (Direct.) 78. The opposite of a proposition is formed by stating the negative of its hypothesis and its conclusion. Thus, If A is equal to B, C is equal to D. (Direct.) 79. The converse of a truth is not necessarily true. Thus, every horse is a quadruped is a true proposition, but the converse, Every quadruped is a horse, is not true. 80. If a direct proposition and its converse are true, the opposite proposition is true; and if a direct proposition and its opposite are true, the converse proposition is true. Let it be granted 1. That a straight line can be drawn from any one point to any other point. 2. That a straight line can be produced to any distance, or can be terminated at any point. 3. That a circumference may be described about any point as a centre with a radius of given length. 1. Things which are equal to the same thing are equal to each other. 2. If equals are added to equals the sums are equal. are 3. If equals are taken from equals the remainders are equal. 4. If equals are added to unequals the sums unequal, and the greater sum is obtained from the greater magnitude. 5. If equals are taken from unequals the remainders are unequal, and the greater remainder is obtained from the greater magnitude. 6. Things which are double the same thing, or equal things, are equal to each other. 7. Things which are halves of the same thing, or of equal things, are equal to each other. 8. The whole is greater than any of its parts. 9. The whole is equal to all its parts taken together. 83. SYMBOLS AND ABBREVIATIONS. + increased by. diminished by. x multiplied by. ; divided by. = is (or are) equal to. is (or are) equivalent to. is (or are) greater than. < is (or are) less than. ... therefore. L angle angles. I perpendicular. Is perpendiculars. Def... definition. right. || parallel. llo parallels. A triangle. A triangles. parallelogram. s parallelograms. O circle. © circles. straight. Q. E. D. . . quod erat demonstran dum, which was to be proved. Q. E. F... quod erat faciendum, which was to be done. BOOK I. THEOREMS. 84. All straight angles are equal. 86. Cor. 2. The angular units, degree, minute, and second, have constant values. 87. Cor. 3. The complements of equal angles are equal. 88. Cor. 4. The supplements of equal angles are equal. 89. Cor. 5. At a given point in a given straight line one perpendicular, and only one, can be erected. 90. If two adjacent angles have their exterior sides in a straight line, these angles are supplements of each other. 92. Cor. Since the angular magnitude about a point is neither increased nor diminished by the number of lines which radiate from the point, it follows that, The sum of all the angles about a point in a plane is equal to two straight angles, or four right angles; The sum of all the angles about a point on the same side of a straight line passing through the point is equal to a straight angle, or two right angles. 93. If two adjacent angles are supplements of each other, their exterior sides lie in the same straight line. 94. Since Theorems 90 and 93 are true, their opposites are true ($ 80); namely: If the exterior sides of two adjacent angles are not in a straight line, these angles are not supplements of each other. If two adjacent angles are not supplements of each other, their exterior sides are not in the same straight line. 95. If one straight line intersects another straight line, the vertical angles are equal. 96. Cor. If one of the four angles formed by the intersection of two straight lines is a right angle, the other three angles are right angles. 97. From a point without a straight line one perpendicular, and only one, can be drawn to this line. 100. Two straight lines in the same plane perpendicular to the same straight line are parallel. 101. Through a given point, one straight line, and only one, can be drawn parallel to a given straight line. 102. If a straight line is perpendicular to one of two parallel lines, it is perpendicular to the other. 104. If two parallel straight lines are cut by a third straight line, the alternate-interior angles are equal. 105. When two straight lines are cut by a third straight line, if the alternate-interior angles are equal, the two straight lines are parallel. 106. If two parallel lines are cut by a third straight line, the exterior-interior angles are equal. 107. Cor. The alternate-exterior angles are equal. 108. When two straight lines are cut by a third straight line, if the exterior-interior angles are equal, these two straight lines are parallel. 109. If two parallel lines are cut by a third straight line, the sum of the two interior angles on the same side of the transversal is equal to two right angles. 110. When two straight lines are cut by a third straight line, if the two interior angles on the same side of the transversal are together equal to two right angles, then the two straight lines are parallel. 111. Two straight lines which are parallel to a third straight line are parallel to each other. 112. Two angles whose sides are parallel, each to each, are either equal or supplementary. 113. Two angles whose sides are perpendicular, each to each, are either equal or supplementary. 114. The perpendicular is the shortest line that can be drawn from a point to a straight line. 116. Two oblique lines drawn from a point in a perpendicular to a given line, cutting off equal distances from the foot of the perpendicular, are equal. 117. Cor. Two oblique lines drawn from a point in a perpendicular to a given line, cutting off equal distances from the foot of the perpendicular, make equal angles with the given line, and also with the perpendicular. 118. The sum of two lines drawņ from a point to the extremities of a straight line is greater than the sum of two other lines similarly drawn, but included by them. 119. Of two oblique lines drawn from the same point in a perpendicular, cutting off unequal distances from the foot of the perpendicular, the more remote is the greater. 120. Cor. Only two equal straight lines can be drawn from a point to a straight line; and of two unequal lines, the greater cuts off the greater distance from the foot of the perpendicular. 121. Two equal oblique lines, drawn from a point in a perpendicular, cut off equal distances from its foot. 122. Every point in the perpendicular, erected at the middle of a given straight line, is equidistant from the extremities of the line, and every point not in the perpendicular is unequally distant from its extremities. 126. Cor. The locus of a point equidistant from the extremities of a straight line is the perpendicular bisector of that line. 137. The sum of two sides of a triangle is greater than the third side, and their difference is less than the third side. 138. The sum of the three angles of a triangle is equal to two right angles. |