81. The necessity for proof. Some theorems seem evident by merely looking at the figure, and the student will doubtless think a proof unnecessary. The eye, however, cannot always detect error, and reasoning enables us to be sure of our conclusions. The danger of trusting the eye is illustrated in the following exercises.* Ex. 43. In the diagrams given below, tell which line of each pair is the longer, a or b, and verify your answer by careful measurement. Ex. 44. In the figures below, are the lines every where the same distance apart? Verify your answer by using a ruler or a slip of paper. Ex. 45. In the figures below, tell which lines are prolongations of other lines. Verify your answers. These diagrams are taken by permission from the Report of the Committee on Geometry, Proceedings of the Eighth Meeting of the Central Association of Science and Mathematics Teachers. POLYGONS. TRIANGLES 82. Def. A line on a plane is said to be closed if it separates a finite portion of the plane from the remaining portion. 83. Def. A plane closed figure is a plane figure composed of a closed line and the finite portion of the plane bounded by it. 84. Def. A polygon is a plane closed figure whose boundary is composed of straight lines only. The points of intersection of the lines are the vertices of the polygon, and the segments of the boundary lines included between adjacent vertices are the sides of the polygon. 85. Def. The sum of the sides of a polygon is its perimeter. 86. Def. Any angle formed by two consecutive sides and found on the right in passing clockwise around the perimeter of a polygon is called an interior angle of the polygon, or, for brevity, an angle of the polygon. In Fig. 1, tion and the adjacent side is called an exterior angle of the polygon. In Fig. 1, 1, 2, 3, 4, and 5 are exterior angles. 88. Def. A line joining any two non-adjacent vertices of a polygon is called a diagonal; as AC, Fig. 1. 89. Def. A polygon which has all of its sides equal is an equilateral polygon. 90. Def. A polygon which has all of its angles equal is an equiangular polygon. 91. Def. A polygon which is both equilateral and equiangular is a regular polygon. 92. Def. A polygon of three sides is called a triangle; one of four sides, a quadrilateral; one of five sides, a pentagon; one of six sides, a hexagon; and so on. TRIANGLES CLASSIFIED WITH RESPECT TO SIDES 93. Def. A triangle having no two sides equal is a scalene triangle. 94. Def. A triangle having two sides equal is an isosceles triangle. The equal sides are spoken of as the sides of the triangle. The angle between the equal sides is the vertex angle, and the side opposite the vertex angle is called the base. 95. Def. A triangle having its three sides equal is an equilateral triangle. Scalene Isosceles Equilateral TRIANGLES CLASSIFIED WITH RESPECT TO ANGLES 96. Def. A right triangle is a triangle which has a right angle. The side opposite the right angle is called the hypoteThe other two sides are spoken of as the sides † of the nuse. triangle. 97. Def. An obtuse triangle is a triangle which has an obtuse angle. 98. Def. An acute triangle is a triangle in which all the angles are acute. * The equal sides are sometimes called the arms of the isosceles triangle. This term will be used occasionally in the exercises. + Sometimes the sides of a right triangle including the right angle are called the arms of the triangle. This term will be found in the exercises. Ex. 46. Draw a scalene triangle freehand: (a) with all its angles acute and with its shortest side horizontal; (b) with one right angle. Ex. 47. Draw an isosceles triangle: (a) with one of its arms horizontal and one of its angles a right angle; (b) with one angle obtuse. 99. Def. The side upon which a polygon is supposed to stand is usually called its base; however, since a polygon may be supposed to stand upon any one of its sides, any side may be considered as its base. The angle opposite the base of a triangle is the vertex angle, and the vertex of the angle is called the vertex of the triangle. K E 100. Def. The altitude of a triangle is the perpendicular to its base from the opposite vertex. In general any side of a triangle may be considered as its base. Thus in triangle EFG, if FG is taken as base, EH is the altitude; if GE is taken as base, FK will be the altitude; if EF is taken as base, the third altitude can be drawn. Thus every triangle has three altitudes. G e H FIG. 1. It will be proved later that one perpendicular, and only one, can be drawn from a point to a line. 101. The sides of a triangle are often designated by the small letters corresponding to the capitals at the opposite vertices; as, sides e, f, and g, Fig. 1. Ex. 48. Draw an acute triangle; draw its three altitudes freehand. Do they seem to meet in a point? Where is this point located? Ex. 49. Draw an obtuse triangle; draw its three altitudes freehand. Do they meet in a point? Where is this point located? Ex. 50. Where do the three altitudes of a right triangle meet? 102. Def. The medians of a triangle are the lines from the vertices of the triangle to the mid-points of the opposite sides. 103. Superposition. When certain parts of two figures are given equal, we can determine by a process of pure reason whether the two figures may be made to coincide. This process is far more accurate than the actual transference of figures, for we are free from physical errors such as have been referred to in § 81. C. Problem. Given line AB less than CD. Apply AB to CD. A C Make AB collinear with CD and let B fall toward D. Then B will fall between C and D, because AB < CD. Question. Under what hypothesis would B fall on D? Problem. Given angle ABC less than angle RST. ABC to angle RST. Solution. Place point B upon point S and make BA collinear with SR. Then BC will fall between SR and ST, because ABC << RST. B C نے A S B D beyond D? Apply angle T R Ex. 51. Solve the problem above by first making BC collinear with ST. Under what hypothesis would BA fall on SR? outside of angle RST? within angle RST? Illustrate each answer by a diagram. Can you choose where BA will fall after you have put BC on ST? Could you have chosen where BA should fall at first? 104. Note. In applying one figure to another, always begin with a Place one end of it on one end of another line and make the two lines collinear on the same side of the point. line. Throughout the process, determine first the direction each line will take, and then where its end will fall. If two lines are given equal, one may be placed upon the other, end on end, for the lines will coincide. Ex. 52. Given two triangles ABC and DEF, such that: (1) AC= DF, angle A is greater than angle D, angle C is less than angle F; (2) AC DF, angle A angle D, angle C is less than angle F; (3) AC = DF, angle A = angle D, angle C = angle F. Apply triangle ABC to triangle DEF and draw a diagram to illustrate each case. = |