BOOK I RECTILINEAR FIGURES 58. A Rectilinear Figure is a geometrical figure composed of straight lines only. 59. Congruence. If asked to compare two sheets of paper as to shape and size, it is natural to place one upon the other to determine whether they can be made to coincide (fit together). 60. Two geometrical figures are congruent (~) if they can be made to coincide. 61. Ax. 15. Congruence Axiom. - Two figures which are congruent to the same figure are congruent to each other. Historical Note. - The symbol was introduced by a mathematician, Leibnitz, in 1679. 62. Superposition is the process of placing one geometrical figure upon another for the purpose of comparing them. Literally, superpose means "place above." Postulate. ·A geometrical figure may be moved about in space without changing any of its parts. Ex. 1. Notice the panes of glass in the windows of your schoolroom. Do they appear to be congruent? Do they coincide now? Ex. 2. Draw ▲ ABC, having AB = 4 in., BC = 6 in., and ▲ B = 50°. (a) Measure Z A, Z C, and AC. (b) Cut your triangle from the paper. Compare it by superposition with the triangles made by other members of your class. (c) What do you conclude must be true about all triangles made according to the directions? Ex. 3. Are the statements of the hypothesis assumed to be true or must they be proved to be true? Answer the same question for the conclusion. PROPOSITION I. THEOREM 63. If two triangles have two sides and the included angle of one equal respectively to two sides and the included angle of the other, the triangles are congruent. Proof. 1. Place ▲ ABC on ▲ DEF with point A on point [Only one st. line can be drawn through two points.] Ax. 10; § 51 6. .. ▲ ABC coincides with ▲ DEF and is congruent to it. are congruent if they can be made to coincide.] [Two $60 Ex. 4. After you place ▲ ABC so that A falls on D, does AB fall on DE, or must you place AB on DE? Ex. 5. Where would C fall if AC were equal to DF? Would the triangles be congruent ? Ex. 6. Where would AC fall if ZA were less than D? Ex. 7. Make a free-hand drawing to illustrate the result if a ▲ ABC is superposed on a ▲ DEF, when AC=} DF, Z A=2D, and AB=} DE. Note. The symbol is read "is identically equal to." No other authority is required, for any magnitude is equal to itself. 65. Homologous parts of congruent figures are parts which are similarly located; they are the parts which coincide when the figures are made to coincide. It follows that: homologous parts of congruent figures are equal. Thus, in § 63, ZC is homologous to F, and side BC is homologous to side EF. It follows that ▲ C F and BC = EF. Note. = - In congruent triangles, homologous sides lie opposite equal angles and homologous angles lie opposite equal sides. 66. Principle I. To prove that two segments are equal or two angles are equal, try to prove them homologous parts of congruent triangles. Note. [Homologous sides of cong. A are equal.] AO lies opposite 21 and OD lies opposite 2; hence they are homologous sides of the congruent triangles. Ex. 15. To obtain the distance AB. (1) Locate point O from which OA and OB may be measured. (2) Extend 40 and BO, making OC=AO and OD: BO. Then DC AB. Prove it. Ex. 16. If AB and CD are two diameters of a circle, prove that AD must equal BC. prove Ex. 17. If, in the adjoining figure, ≤3=27, 2=27. Ex. 18. If 24 = 25, prove 21 = 28. 7/8 Ex. 19. If ≤3=26, prove ≤1= 25. Suggestions. -1. Recall § 41. 2. Of what angle is 21 the supplement? 3. Of what angle is 5 the supplement? Ex. 20. If 24 = 28, prove ≤3 = 26. Ex. 21. When are two figures congruent ? Ex. 22. What method of proof is employed in proving Proposition I? Ex. 23. Draw a ▲ ABC, having AB-4 in., ZA 60°, and ▲ B=80°. Cut your triangle from the paper and compare it with the triangles made by other members of your class. What do you conclude must be true about all triangles made according to the directions given? |