PLANE GEOMETRY. INTRODUCTORY DEFINITIONS. 1. Mathematics is the science of quantity. 2. Quantity is that which can be measured; as distance, time, weight. 3. Geometry is that branch of mathematics which treats of the properties of extension. 4. Extension has one or more of the three dimensions, length, breadth, or thickness. 5. A Point has position, but not magnitude. 6. A Line has length, without breadth or thickness. 7 A Straight Line is one whose direction is the same throughout; as A B. A straight line has two directions exactly opposite, of which either may be assumed as its direction. The word line, used alone in this book, means a straight line. 8. Corollary. Two points of a line determine its position. 9. A Curved Line is one whose direction is constantly changing; as CD. C 10. A Surface has length and breadth, but no thickness. D 11. A Plane is such a surface that a straight line joining any two of its points is wholly in the surface. 12. A Solid has length, breadth, and thickness. 13. Scholium. The boundaries of solids are surfaces; of surfaces, lines; the ends of lines are points. 14. A Theorem is something to be proved. 15. A Problem is something to be done. 16. A Proposition is either a theorem or a problem. 17. A Corollary is an inference from a proposition or state ment. 18. A Scholium is a remark appended to a proposition. 19. An Hypothesis is a supposition in the statement of a proposition, or in the course of a demonstration. 20. An Axiom is a self-evident truth. AXIOMS. 1. If equals are added to equals, the sums are equal. 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. 5. Like powers and like roots of equals are equal. 6. The whole of a magnitude is greater than any of its parts. 7. The whole of a magnitude is equal to the sum of all its parts. 8. Magnitudes respectively equal to the same magnitude are equal to each other. A straight line is the shortest distance between two points. BOOK I. ANGLES, LINES, POLYGONS. ANGLES. DEFINITIONS. 1. An Angle is the difference in direction of two lines. If the lines meet, the point of meeting, B, is called the vertex; and the lines A B, B C, the sides of the angle. B If there is but one angle, it can be designated by the letter at its vertex, as the angle B; but when a number of angles have the same vertex, each angle is designated by three letters, the middle letter showing the vertex, and the other two with the middle letter the sides; as the angle A B C. 2. If a straight line meets another so as to make the adjacent angles equal, each of these angles is a right angle; and the two lines are perpendicular to each other. Thus, ACD and D C B, being equal, are right angles, and AB and DC are perpendicular to each other. A D B 3. An Acute Angle is less than a right angle; as EC B. 4. An Obtuse Angle is greater than a right 4. angle; as ACE. Acute and obtuse angles are called oblique angles. E B 5. The Complement of an angle is a right angle minus the given angle. Thus (Fig. in Art. 7), the complement of A CD is ACF ACD DCF. 6. The Supplement of an angle is two right angles minus the given angle. Thus (Fig. Art. 7), the supplement of A CD is (A CFFCB)-ACD=DCB. THEOREM I. 7. The sum of all the angles formed at a point on one side of a straight line, in the same plane, is equal to two right angles. Let DC and E C meet the straight line AB at the point C; then ACD+DCE + ECB = two right angles. At C erect the perpendicular, CF; then it is evident that F D E A B C ACD+DCE+ECB ACD+DCF+FCE+ECB = = ACF+FCB=two right angles. 8. Corollary 1. If only two angles are formed, each is the supplement of the other. For by the theorem, ACD+DCB= two right angles; therefore AC D=two right angles-DCB, or DCB two right angles-A CD. = 9. Corollary 2. The sum of all the angles formed in a plane about a point is equal to four right angles. Let the angles ABD, DBE, EBF, FBG, GBA, be formed in the same plane about the point B. Produce AB; then the sum of the angles above the line AC is equal to two right angles; and also, the sum of the angles below the line AC is equal A D E B to two angles (7)*; therefore the sum of all the angles at the point B is equal to four right angles. THEOREM II. 10. If at a point in a straight line two other straight lines upon opposite sides of it make the sum of the adjacent angles equal to two right angles, these two lines form a straight line. Let the straight line D B meet the two lines, A B, BC, so as to make ABD+DBC= two right angles : then AB and BC form a straight line. A B D E C For if A B and B C do not form a straight line, draw BE so that A B and B E shall form a straight line; then ABD DBE two right angles (7); but by hypothesis, therefore ABD+DBC= two right angles; DBE DBC the part equal to the whole, which is absurd (Axiom 6); therefore A B and B C form a straight line. THEOREM III. 11. If two straight lines cut each other, the vertical angles are equal. Let the two lines, AB, CD, cut each other at E; then AEC DE B. * The figures alone refer to an article in the same Book; in referring to an article in another Book the number of the Book is prefixed. |