REVIEW QUESTIONS 1 What is a proposition? 2 How are propositions distinguished? Define each. 3 What is a demonstration ? a solution? 4 Distinguish between a direct demonstration and an indirect demonstration. 5 Distinguish between the two parts of a theorem. 6 What is the converse of a theorem? the opposite? 7 State the logical relation existing between a theorem, its converse, and its opposite. 8 Define magnitude; extension; space. 9 Distinguish between a material solid and a geometric solid. 10 How many dimensions has a solid ? a surface? a line? a point? 11 What are the boundaries of a solid ? of a surface? of a line? 12 How may a line be traced? a surface? a solid ? 13 Give the abstract conception of a point; of a line; of a surface; of a solid. 14 What are the fundamental concepts in Geometry? Define each. 15 Give the classification of lines and define each. 16 Define a plane surface; a curved surface. 17 Define Geometry (§ 56). 18 Give the classification of angles, and define each. 19 What is the complement of an angle? the supplement? 20 What are perpendicular lines? oblique lines? 21 Define geometric figure; plane figure; rectilinear figure; curvilinear figure. 22 What is the unit of angular measurement? How obtained ? 23 What are the geometric magnitudes ? 24 Distinguish between similar, equivalent, and equal magnitudes. 25 What is the test of equality of geometric magnitudes ? 26 What is the object of Geometry? 27 How is Geometry divided ? 28 Of what does Plane Geometry treat? 29 Can you prove an axiom ? 30 Distinguish between an axiom and a postulate. 31 Can you give an axiom or a postulate not found in the text? 32 How many points determine a straight line? 33 What are supplementary adjacent angles? vertical angles ? 34 What are parallel lines? 35 Define a polygon; the sides, angles, and vertices of a polygon. 36 What are adjacent angles of a polygon? 37 What is an exterior angle of a polygon? 38 Give the classification of triangles, and define each. 39 Define an equiangular triangle; an equilateral triangle. 40 Is an equilateral triangle equiangular? why ? 41 Is an equiangular triangle equilateral ? why ? 42 Can a leg of a right triangle be equal to the hypotenuse? 43 What is the base of a triangle ? 44 What is the base of an isosceles triangle? 45 What is the vertex of a triangle ? 46 Define a median, an altitude, an angle-bisector of a triangle. 47 Define mutually equilateral polygons; mutually equiangular polygons. 48 What are homologous lines in mutually equiangular polygons? 49 Define quadrilateral; trapezium; trapezoid. 50 What does the word "parallelogram" mean? 51 Define rectangle; square; rhomboid; rhombus. 52 Define the bases of a trapezoid; the median; the legs. 53 What is the altitude of a parallelogram or trapezoid? 54 What does the word "polygon" mean? 55 Define pentagon; hexagon; heptagon; octagon; decagon; dodecagon; pentadecagon. 56 Define equilateral polygon; equiangular polygon; convex polygon; concave polygon. 57 How may any polygon be divided into triangles? 58 Write a short essay on the great geometers Thales and Pythagoras. SUGGESTIONS ON THE TREATMENT OF EXERCISES 219 In Arithmetic and Algebra definite rules are given for the solution of problems. No such specific directions can be given for the treatment of original exercises in Geometry. The following general directions, however, will greatly assist the beginner: 1 Draw general figures. Thus, if you are dealing with a triangle in general, draw a scalene triangle; if with a parallelogram, draw a rhomboid. Observe that this is always done in the text. 2 Draw figures accurately. An accurate figure frequently suggests a clue to the proof; an inaccurate figure leads into error. 3 Fix clearly in mind the things given about the figure, and the precise thing in the figure to be proved. This direction is fundamental; its disregard is the prime cause of the beginner's frequent failure. Observe under "Hypothesis" and "Conclusion" how carefully and clearly this injunction is observed in every proposition of the text. 4 In the proof make use of every condition given in the hypothesis. Work along lines involving only a part of the things given will end in certain failure. No relation would be given were it not a necessary part upon which to build the proof. Examine the proof of any proposition in the text, and observe that in such proof every condition given in the hypothesis is always used. 5 Begin by supposing the theorem true; then, step by step, trace out the relations which follow this supposition, until some known theorem is reached. Now begin with this known theorem and arrange the steps in reverse order, thus leading up to the required proof. In illustration consider the following THEOREM. The bisectors of two adjacent angles of a parallelogram are perpendicular to each other. In the DEFG let DH and EK bisect the D and E respectively. We are to prove that c is a rt. angle. Suppose that c is a rt. Z. Then a + 20 B = a rt. ≤ (§ 155). Then ≤ D+ ≤ E = 2 rt. K F (Ax. 7). But this last conclusion is a known theorem (§ 124) with which we begin the direct This is the usual method of attacking most exercises in Geometry, and is called Analysis. The reverse process, the orderly building up from known facts until a new truth is obtained, is called Synthesis. It should be observed that the synthetic method of proof is used in most text propositions. 6 When other means fail, especially in converse theorems, try the indirect method. See §§ 115, 116. 7 Draw such auxiliary lines as may be necessary to give a clue to the line of argument. See the dotted lines in the figures of many of the text propositions; also, Exs. 138, 150, 151. The ability to attack exercises successfully and rapidly implies a comprehensive knowledge of the chief propositions of the text, and a keen recognition of their application at sight. To this end we here remind the student of some of the cardinal truths which should ever be vividly in his mind. Two lines are equal: 1 If they are homologous sides of equal triangles. § 166 § 176 § 196 § 200 Two angles are equal: 1 If they are complements or supplements of equal angles. §§ 104, 105 § 112 2 If they are vertical angles. 3 If they are alternate-interior or corresponding angles of parallel lines. §§ 121, 122 4 If they are homologous angles of equal triangles. 5 If they are opposite equal sides in a triangle. § 166 § 173 6 If they are the opposite angles of a parallelogram. § 203 Two triangles are equal: 1 If two sides and the included angle of the one are equal respectively to two sides and the included angle of the other. § 162 2 If two angles and the included side of the one are equal respectively to two angles and the included side of the other. § 167 3 If the three sides of the one are equal respectively to the three sides of the other. § 172 Two right triangles are equal: 1 If their legs are equal, each to each. § 163 2 If the hypotenuse and an acute angle of the one are equal respectively to the hypotenuse and an acute angle of the other. § 168 3 If a leg and an acute angle of the one are equal respectively to a leg and the homologous acute angle of the other. § 169 4 If the hypotenuse and a leg of the one are equal respectively to the hypotenuse and a leg of the other. Two lines are parallel : § 178 1 If they are perpendicular to the same straight line. § 115 2 If they are parallel to a third straight line. § 118 |