APPENDIX. BEFORE attempting the exercises appended, it will be well to consider the various methods of proof adopted in the preceding lessons. Theorems may be demonstrated by I. The indirect proof, with reductio ad absurdum. This is generally used in demonstrating converse propositions, and is not often required in exercises. Either no construction is required, as in I. 19, or it is very simple, and follows at once from the assumption that the theorem is not true, e.g. I. 6, 14, 26. II. The method of superposition. This is fully described in Lecture IV. I. 8 is a simple example of its use. III. The method of comparison. A very large number of theorems require the equality, or inequality, of certain lines and angles to be established. This can often be effected by comparing them with other lines and angles whose relative size we know, either by means of the angles or lines already existing in the figure, as in I. 15, or more often by constructing others, which serve as a medium of comparison between those whose equality, or inequality, we wish to show. I. 17, 18, and 21, are good examples of this method. IV. The method of analysis. A clue to the more difficult constructions can be obtained by this method, for example I. 16. (See Lecture VI.) Problems may be solved by the method of loci, as I. 22, or by analysis, as I. 9, &c. A problem is said to be determinate when, with the data, it admits of one definite solution. Thus, in I. 22, only one triangle can be constructed satisfying the required conditions. A problem is indeterminate when it admits of more than one definite solution, e.g. if it be required to construct a triangle equal in area to a given triangle, we know that any triangle on an equal base and between the same parallels as the given triangle will satisfy the required conditions (I. 38). Sometimes certain 'limitations' exist, within which alone the solution of a problem is possible, e.g. in I. 22, of the three given lines any two must be greater than the third. EXERCISES. BOOK I. LECTURE III. (PROPOSITIONS 1-3). 1. On a given base describe an isosceles triangle, whose equal sides shall each be equal to a given straight line. 2. Draw figures illustrating the following cases of I. 2 :— (a) When A falls in BC produced. (B) When points A and B coincide. (y) When A falls within BC. 3. In the figure of I. 2, if a side of the equilateral triangle is equal to the given line, show that the radius of the larger circle is equal to the diameter of the smaller. 4. Draw a straight line three times as long as a given straight line. 5. The shorter diagonal being given, construct a rhombus whose sides shall each be equal to the given diagonal. 6. Show that the two triangles into which the longer diagonal divides a rhombus are isosceles. 7. Construct an isosceles triangle, each of whose equal sides is double of the base. 8. Produce the shorter of two given straight lines so that it may be equal to the longer. 9. Construct the figure of I. 2 by producing BD and AD through the vertex of the equilateral triangle, and prove. 10. In the fig. of I. 1 produce CA, CB to meet the circles in E and F. Join EF, and show that the triangle EFC is isosceles. LECTURE IV. (PROPOSITIONS 4-6). 1. Demonstrate formally the corollary to I. 5. 2. In an isosceles triangle the straight line which bisects the vertical angle also bisects the base. 3. The straight lines bisecting the angles at the base of an isosceles triangle (having base BC) meet at a point O. Show that OBC is an isosceles triangle. 4. Demonstrate formally the corollary to I. 6. 5. Let ABC be an equilateral triangle having its sides bisected in DEF. Show that the straight lines joining these points with the opposite angles are equal. 6. In a triangle ABC the vertical angle A is half of the angle B, which is bisected by the line BD, meeting the opposite side in D. Show that the triangle BDA is isosceles. 7. In the fig. of I. 5 let O be the point at which BG and FC intersect. Show that the triangle OBC is isosceles, and that FO is equal to GO. 8. If two straight lines bisect each other at right angles, any point in either of them is equidistant from the extremities of the other. 9. Distinguish between equal and coincident triangles. 10. If a triangle ABC be turned over about its side AB, show that the line joining the two positions of C is perpendicular to AB. 11. An equiangular quadrilateral which has two adjacent sides equal to each other is also equilateral. 12. Produce BA, CA, sides of an equilateral triangle, through A, making AD, AE equal to BA, CA, each to each. Join BE, DE, DC. What do we know about the figure thus formed; and about the triangles of which it is composed ? 13. Two sides of a triangle being produced, if the angles on the other side of the base be equal, the triangle is isosceles. 14. If the angles of one triangle be equal to the angles of another, each to each, are the triangles necessarily equal? What is the force of the expression, each to each'? Exhibit a case when the angles are respectively equal, but not each to each. LECTURE V. (PROPOSITIONS 7-8). 1. In the figure of I. 7, let the angle ABD be a right angle, and the angle ABC obtuse; demonstrate the proposition under these conditions. 2. The opposite sides of a quadrilateral ABCD are equal: show that the diagonal bisects it. 3. If two triangles have their sides equal, each to each, the triangles are equal in area. 4. ABC and DBC being two triangles, having their three sides equal each to each on a common base, join AD and prove that the triangles ADB, ADC are equal in all respects. 5. If a side of an equilateral triangle be double of the side of another equilateral triangle, what proportion will the two triangles bear to each other? 6. In the figure of I. 5 join O, the point at which BG and FC intersect with A, and show that this line bisects the angle BAC. 7. Show that the straight lines joining the points of bisection of the sides of an equilateral triangle divide the figure into four equilateral triangles. 8. The opposite angles of a rhombus are equal. 9. The diagonals of a rhombus bisect one another and the angles through which they pass. 10. BAC, BDC are isosceles triangles on opposite sides of a common base BC, join AD, and show that it bisects (a) the base, (B) the vertical angles of the triangles, (y) the whole figure. LECTURE VI. (PROPOSITIONS 9-12). 1. In the figure of I. 9, construct the equilateral triangle on the same side as A; demonstrate the case in which the point F falls without A. 2. Produce a given straight line so that the part produced may be one-fifth of the line when produced. 3. In a given straight line find a point which is equidistant from two given points. 4. If two isosceles triangles be described on the same base, the straight line joining their vertices (produced if necessary) bisects the base at right angles. 5. Divide an obtuse angle into four equal parts. 6. On a given base construct an isosceles triangle whose altitude is equal to the base.1 7. With common radius AB, draw two circles intersecting as in the fig. of I. 1, join the points of intersection: it is required to show that these straight lines bisect, and are perpendicular to each other. 1 The altitude of a triangle is the perpendicular let fall from the vertex to the base (produced if necessary). |