5. When two sides of a triangle are unequal, the angle opposite the greater is greater than that opposite the less. 6. An equilateral triangle is equiangular. 7. An equiangular triangle is equilateral. 8. Show how to trisect a given angle. Describe a trisector. 9. Construct a triangle, having given Ist. The three sides. 2nd. Two sides and the included angle. 3rd. Two angles and included side. 4th. Two sides and an angle opposite one. Show when two triangles can be constructed fulfilling the conditions in this case, and when only one. 10. Construct a right-angled triangle, having given Ist. The hypothenuse and an angle adjacent to it. II. Prove by superposing the triangles, that two triangles are equal Ist. When two sides and the included angle of one are respectively equal to the corresponding parts of the other. 2nd. When two angles and the included side of the one are respectively equal to the corresponding parts of the other. 3rd. When the three sides of the one are respectively equal to the three sides of the other. 12. Prove that when two sides of a triangle are respectively equal to two sides of another, but the included angles are unequal, the bases are unequal. Theorems and Problems. 1. The line that joins the vertex to the middle point of the base of a triangle is less than half the sum of the two sides. 2. The perimeter of a triangle is greater than the sum of the lines joining the middle point of the sides with the opposite angles. 3. The perimeter of a triangle is greater than the sum of the straight lines which join any point in the interior of the triangle with the three angles, and is less than twice this sum. 4. If two angles of a triangle be bisected, the perpendiculars from their point of intersection to the sides are equal. 5. If two exterior angles of a triangle be bisected, the perpendiculars from their point of intersection to the sides are equal. 6. From two summer-houses I wish to make two straight walks of the same length to the same point in a distant road : how shall I find the point? 7. Describe an isosceles triangle on a given base, each of whose sides shall be double of the base. 8. Construct a triangle, having given the base, sum of the sides, and one of the angles at the base. 9. Construct a triangle, having given the base, difference of the sides, and one of the angles at the base. 10. Construct a triangle, having given the base the difference of the sides, and difference of the angles at the base. II. Construct an isosceles triangle, having given the vertical angle and a point in the base. 12. Construct a triangle, having given Ist. Two sides, and a line joining the middle point of base with the opposite angle. 2nd. Two sides, and a line joining the middle point of one of them with the opposite angle. 3rd. The lines drawn from the middle points of the sides to the opposite angles. 4th. The feet of the three heights. 5th. One angle, the perimeter, and the height from the angle. 6th. One angle, the perimeter, and the height opposite the angle. 7th. A side, an adjacent angle, and the length of the bisector of this angle. 8th. The sum of two sides and the angles. 9th. The perimeter and the angles. Ioth. An angle, the length of the bisector, and the height from this angle. 11th. Ditto, the height being opposite the angle. 12th. The base, the sum of the two sides, and the difference of the angles at the base. 13th. The perimeter, the size and position of one angle, and a point in the opposite side. 14th. A side, an angle, and a height (five cases). 13. Construct a right-angled triangle, having given— Ist. One of the sides and the excess of the hypothenuse over the other side. 2nd. The angles and the difference of the hypothenuse and one of the other sides. CHAPTER V. PARALLELS AND QUADRILATERALS. Parallels. 91. Two straight lines in the same plane, which do not meet however far they may be produced, are called parallel lines (fig. 80). Such would be two lines drawn on writing paper along the edges of a ruler. It is of course impossible so to produce two parallel straight с Fig. 80. D lines as to be sure they will never meet; but we shall deduce from the above definition, other tests of parallelism which will be applicable in that case. Two straight lines perpendicular to a third are parallel. 92. Let ABCD (fig. 81) be two perpendiculars to the same straight line AC. If the two lines AB, CD meet, then it is possible from their point of intersection to let fall D two perpendiculars upon the straight line A C. But this we know to be impossible (§ 36). Therefore these two straight lines do not meet. C Fig. 81. B 93. Two plumb-lines fixed upon a mason's level are parallel. Fold a sheet of paper as in figure 20, then make a third crease by folding it farther over, keeping the same lines in contact, two parallel lines will be formed perpendicular to a third. It is evident that two straight lines which intersect cannot both be parallel to a given straight line. To draw parallel lines by means of the ruler, the square, and the compasses. 94. We see from the foregoing that the drawing of parallel lines resolves itself into drawing perpendiculars to a given straight line, and therefore the ruler, the square, and the compasses, instruments already described, may be here employed. Suppose, for example, we wish to draw on paper a series of parallel straight lines. Lay a ruler upon the paper, and slide the square along it, so as to draw a series of perpendiculars to the ruler. All these lines will be parallel. Through a given point to draw a straight line parallel to a given straight line. 95. Let A B be the given straight line, and C the given point (fig. 81); from the point C drop the perpendicular CA upon A B, and upon CA at the point C raise the perpendicular CD. CD is parallel to A B. To draw parallel straight lines upon a level piece of ground. B F F' 96. In a similar way to that described in section 95, by using suitable instruments, parallel lines may be drawn upon a plot of ground. Having drawn one of the lines AB (fig. 82), raise a perpendicular to it by the help of a cord as previously explained. Then take a long rod and lay it upon the straight line CD E Fig. 82. E' Ꭰ from C to E, attach a long cord to its two extremities, and stretch the cord until it is tight, and so that one part of it may fall on C B. Mark in the cord the point F. Then CEF will be a large portable square. To draw a parallel to C E, remove the rod to C'E' and stretch the cord at the point F so that both parts are tight, then this point will take the position F', and the straight line C'F' will be parallel to A B, and may be produced as far as we please. Every straight line perpendicular to one of two parallel straight lines, is also perpendicular to the other. 97. Let CA (fig. 81) be perpendicular to AB, then it will also be perpendicular to C D which is parallel to A B. Conceive a line drawn through the point C perpendicular to CA; this line will be parallel to A B (92), and will therefore coincide with CD, since at the point C only one straight line can be drawn parallel to A B (893). Hence CD is perpendicular to CA. Equality of exterior and interior alternate angles and corresponding angles. 98. When a straight line EF meets two parallel straight lines, AB, CD (fig. 83), it makes with them eight angles, which bear special names. The angles HGA and GHD on either side of it, are called the interior alternate angles, as also are the angles H G B, G H C. The angles BGE, CH F are called the exterior alternate angles. The two angles AGH, CHF, on the same side of EF, the one between, the other outside the parallel lines, are said to be corresponding angles. H E G B Fig. 83. t The angles bearing the same name, are equal to one another. Let us demonstrate this in the case of the interior alternate angles HGA, GHD. Find the middle, I, of G H, and from this point draw the perpendicular KL; it will also be perpendicular to A B. Thus two right-angled triangles are formed with the common vertex I. These triangles have the sides IG, I H equal, and the angles at I equal; therefore the triangles are equal, and the third angle of the one is equal to the third angle of the other, namely the interior-alternate angles, I G A to IH D. If to these equal angles the angle I G B on the one |