Angles, like other quantities, are susceptible of addition, subtraction, multiplication, and division; thus, the angle DCE Fig.20. (fig. 20) is the sum of the two angles DCB, BCE, and the angle DCB is the difference between the two angles DCE, BCE. Fig. 3. Fig. 4. Fig. 5. 10. When a straight line AB (fig. 3) meets another straight line CD in such a manner that the adjacent angles BAC, BAD, are equal, each of these angles is called a right angle, and the line AB is said to be perpendicular to CD. 11. Every angle BAC (fig. 4), less than a right angle, is an acute angle; and every angle, DEF, greater than a right angle is an obtuse angle. 12. Two lines are said to be parallel (fig. 5), when, being situated in the same plane and produced ever so far both ways, they do not meet. 13. A plane figure is a plane terminated on all sides by lines. If the lines are straight, the space which they contain is Fig. 6. called a rectilineal figure, or polygon (fig. 6), and the lines taken together make the perimeter of the polygon. Fig. 7. 14. The polygon of three sides is the most simple of these figures, and is called a triangle; that of four sides is called a quadrilateral; that of five sides, a pentagon; that of six, a hexagon, &c. 15. A triangle is denominated equilateral (fig. 7), when the Fig, 8. three sides are equal, isosceles (fig. 8), when two only of its sides Fig. 9. are equal, and scalene (fig. 9), when no two of its sides are equal. 16. A right-angled triangle is that which has one right angle. The side opposite to the right angle is called the hypothenuse. Fig. 10. Thus ABC (fig. 10) is a triangle right-angled at A, and the side BC is the hypothenuse. Fig. 11. Fig. 12. Fig. 13. Fig. 14. Fig. 15. 17. Among quadrilateral figures we distinguish ; The square (fig. 11), which has its sides equal and its angles right angles, (See art. 80); The rectangle (fig. 12), which has its angles right angles without having its sides equal (See art. above referred to); The parallelogram (fig. 13), which has its opposite sides parallel ; The rhombus or lozenge (fig. 14), which has its sides equal without having its angles right angles; The trapezoid (fig. 15), which has two only of its sides parallel. 18. A diagonal is a line which joins the vertices of two angles not adjacent, as AC (fig. 42). 19. An equilateral polygon is one which has all its sides equal; an equiangular polygon is one which has all its angles equal. 20. Two polygons are equilateral with respect to each other, when they have their sides equal, each to each, and placed in the same order, that is, when by proceeding round in the same direction the first in the one is equal to the first in the other, the second in the one to the second in the other, and so on. In a similar sense are to be understood two polygons equiangular with respect to each other. The equal sides in the first case, and the equal angles in the second, are called homologous (A). 21. An Axiom is a proposition, the truth of which is self-evident. A Theorem is a truth which becomes evident by a process of reasoning called a demonstration. A Problem is a question proposed which requires a solution. A Lemma is a subsidiary truth employed in the demonstration of a theorem, or in the solution of a problem. The common name of Proposition is given indifferently to theorems, problems, and lemmas. A Corollary is a consequence which follows from one or several propositions. A Scholium is a remark upon one or more propositions which have gone before, tending to show their connexion, their restriction, their extension, or the manner of their application. A Hypothesis is a supposition made either in the enunciation of a proposition, or in the course of a demonstration. Axioms. 22. Two quantities, each of which is equal to a third, are equal to one another. 23. The whole is greater than its part. 24. The whole is equal to the sum of all its parts. 25. Only one straight line can be drawn between two points. 26. Two magnitudes, whether they be lines, surfaces, or solids, are equal, when, being applied the one to the other, they coincide with each other entirely, that is, when they exactly fill the same space. Fig. 42. PART FIRST. OF PLANE FIGURES. SECTION FIRST. First Principles, or the Properties of perpendicular, oblique, and parallel Lines. THEOREM. 27. ALL right angles are equal. Demonstration. Let the straight line CD be perpendicular to Fig. 16. AB (fig. 16), and GH to EF, the angles ACD, EGH, will be equal. Take the four distances CA, CB, GE, GF, equal to each other, the distance AB will be equal to the distance EF, and the line EF may be applied to AB, so that the point E will fall upon A, and the point F upon B. These two lines, thus placed, will coincide with each other throughout; otherwise there would be two straight lines between A and B, which is impossible (25). The point G therefore, the middle of EF, will fall upon the point C, the middle of AB. The side GE being thus applied to CA, the side GH will fall upon CD; for, let us suppose, if it be possible, that it falls upon a line CK, different from CD; since, by hypothesis (10), the angle EGH = HGF, it follows that Fig. 17. But and besides, by hypothesis, hence and the line GH cannot fall upon a line CK different from CD ; consequently it falls upon CD, and the angle EGH upon ACD, and EGH is equal to ACD; therefore all right angles are equal. THEOREM. 28. A straight line CD (fig. 17), which meets another straight line AB, makes with it two adjacent angles ACD, BCD, which, taken together, are equal to two right angles. Demonstration. At the point C, let CE be perpendicular to AB. The angle ACD is the sum of the angles ACE, ECD; therefore ACD + BCD is the sum of the three angles ACE, ECD, BCD. The first of these is a right angle, and the two others are together equal to a right angle; therefore the sum of the two angles ACD, BCD, is equal to two right angles. 29. Corollary 1. If one of the angles ACD, BCD, is a right angle, the other is also a right angle. 30. Corollary 11. If the line DE (fig. 18) is perpendicular to Fig. 18. AB; reciprocally, AB is also perpendicular to DE. For, since DE is perpendicular to AB, it follows that the angle ACD is equal to its adjacent angle DCB, and that they are both right angles. But, since the angle ACD is a right angle, it follows that its adjacent angle ACE is also a right angle; therefore the angle ACE = ACD, and AB is perpendicular to DE. 31. Corollary 111. All the successive angles, BAC, CAD, DAE, EAF, (fig. 34), formed on the same side of the straight Fig. 34. line BF, are together equal to two right angles; for their sum is equal to that of the two angles BAM, MAF; AM being perpendicular to BF. THEOREM. 32. Two straight lines, which have two points common, coincide throughout, and form one and the same straight line. Demonstration. Let the two points, which are common to the two lines, be A and B (fig. 19). In the first place it is evident that they must coincide entirely between A and B ; otherwise, two straight lines could be drawn from A to B, which is impossible (25). Now let us suppose, if it be possible, that the lines, when produced, separate from each other at a point C, the one becoming CD, and the other CE. At the point C, let CF be. drawn, so as to make the angle ACF, a right angle; then, ACD being a straight line, the angle FCD is a right angle (29); and, because ACE is a straight line, the angle FCE is a right angle. But the part FCE cannot be equal to the whole FCD; whence straight lines, which have two points common A and B, cannot separate the one from the other, when produced; therefore they must form one and the same straight line. Fig. 19. Fig. 20. THEOREM. 33. If two adjacent angles ACD, DCB (fig. 20), are together equal to two right angles, the two exterior sides AC, CB, are in the same straight line. Demonstration. For if CB is not the line AC produced, let CE be that line produced; then, ACE being a straight line, the angles ACD, DCE, are together equal to two right angles (28); but, by hypothesis, the angles ACD, DCB, are together equal to two right angles, therefore ACD+DCB=ACD+DCE. Take away the common angle ACD, and there will remain the part DCB equal to the whole DCE, which is impossible; therefore CB is the line AC produced. Fig. 21. Fig. 22. THEOREM. 34. Whenever two straight lines AB, DE (fig. 21), cut each other, the angles opposite† to each other at the vertex are equal. Demonstration. Since DE is a straight line, the sum of the angles ACD, ACE, is equal to two right angles; and, since AB is a straight line, the sum of the angles ACE, BCE, is equal to two right angles; therefore ACD+ACE=ACE+BCE; from each of these take away the common angle ACE, and there will remain the angle ACD equal to its opposite angle BCE. It may be demonstrated, in like manner, that the angle ACE is equal to its opposite angle BCD. 35. Scholium. The four angles, formed about a point by two straight lines which cut each other, are together equal to four right angles; for the angles ACE, BCE, taken together, are equal to two right angles; also the other angles ACD, BCD, are together equal to two right angles. In general, if any number of straight lines, as CA, CB (fig. 22), &c., meet in the same point C, the sum of all the successive angles, ACB, BCD, DCE, ECF, FCA, will be equal to four right angles. For, if at the point C, four right angles be formed by two lines perpendicular to each other, they will comprehend the same space as the successive angles, ACB, BCD, &c. These are often called vertical angles. |