54. C. As aA aD, and aB = aE, aC ~ aF? = B.) Triangles agreeing in two angles and the side opposite to one of them are identical. Place tABC upon tABD with AB upon AB. As aABC = aABD, where will BC fall? From A draw AEL BD. 86.3. (1.) Let AC (= AD) be not less than AB. Then can either AC or AD fall between AB and AE? Must, then, AC and AD both fall upon the same side of AE? As AC is equal to AD, can it fall anywhere else except upon AD? 86, 90. Where must C fall? tABCtABD? (2.) Let AC (= AD) be < AB. Can AC or AD now fall between AB and AE? 17. a. XL. A 86. B C E D 86, 90. If both AC and AD fall upon the same side of AE, must they form the same line? Are the given triangles in this case identical? If AC and AD fall upon opposite sides of AE, as in the figure above, then tABC As AC tABD? = AD, what kind of a triangle forms the difference between tABD and tABC? What kind of an angle is ACB? ADB? If AC and AD should be = AE, where would C and D then fall? 66, 89. Would the triangles in this case be identical? What kind of triangles would they be? C.) Two triangles agreeing in two sides and the angle opposite to one of them are identical; except when the given angle is opposite to the less of the two given sides, and the angle apposite to the greater side is obtuse in one of the triangles and acute in the other. a.) Can a right angle ever be opposite to the less of two sides of a triangle? 75. e. Can, then, the uncertainty above ever arise in a right triangle? Right triangles agreeing in any two like sides (§ 58) are identical. b.) Can the uncertainty ever arise, when the given angle is obtuse? § 101. We have now considered (§§ 39, 40, 98–100) all the cases in which triangles agree in three similarly situated and independent parts (meaning by independent parts those which are not determined by others that are given, and of course excluding from the number the third angle, as it is determined by the other two, § 54). The result may be expressed in the following general theorem, which, however, must be received with the qualification above stated (§ 100. C). THEOR. XIII. Triangles agreeing in any three similarly situated and independent parts are identical. § 102. a.) If triangles agree in the three angles, can you determine from this that they are identical? If simply the three angles of a triangle are given, can you determine any thing in respect to the size of the triangle. b.) The following direct mode of proof might have been employed in § 98. XLI. A In the triangles ABC and DBC, let AB = DB, AC = CD, and BC (a side not less than either of the others) = BC. Unite the triangles as in the figure, and join AD. Show (as BC is not less B than AB or AC), that AD must fall between B and C. aBAD aBDA? C 69. aCAD aCDA? aBAC aBDC? tBACtBDC? 20. d. § 103. DEFINITION. A quadrilateral of which the opposite sides are parallel is termed a PARALLELOGRAM'. PROPOSITION XIV. § 104. I. Given, any parallelogram ABCD. Required, AB≈≈≈ DC, AD ≈≈ BC, aBaD, and aA≈≈≈ aC. Join AC. Then, as AB || DC, aBAC ≈≈≈ aDCA? As BC || AD, aBCA aDAC? .. As AC is common, tBAC≈ tDCA? AB DC, AD≈≈≈ BC, and aBaD? C 46. C. And aA (= BAC+DAC) ≈≈≈ aC (= DCA + BCA)? 20. d. § 105. II. Given, in the quadrilateral ABCD, AB= DC, and AD = BC. Required, AB DC, and ADBC? D XLII. (1) Gr. παραλληλόγραμμος, bounded by parallel lines, from παράλληλος, parallel, and rezuμn, line. Join AC. Can you then show that the triangles BAC and DCA are identical? And .. aBAC aDCA? AB } DC? § 106. III. Given, in the quadrilateral ABCD, aA = aC, and aD =aB. Required, AB # DC, and AD BC. And 98. 49. C. XLIII. A B D C aAaB = hm L? .. AD BC? § 107. IV. Given, in the quadrilateral ABCD (Fig. XLII.), AB = and || DC. Required, BC and AD. Join AC. Then, as AB || DC, ɑBAC ≈≈ aDCA? 46. C. 40. § 108. THEOR. XIV. (1.) The opposite sides and angles of a parallelogram are equal; and (2.) any quadrilateral of which the opposite sides or (3.) the opposite angles are equal, or (4.) of which two sides are equal and parallel, is a parallelogram. How many pairs of equal parts does each parallelogram contain? § 109. a.) The two angles of a parallelogram adjacent to any side hm L? = = hm3 ? = 46. e. If, then, one angle of a parallelogram is a right angle, what must the other angles be? If one is acute? If one is obtuse? Must a parallelogram be either right-angled throughout, or oblique-angled throughout? If any two adjoining sides of a parallelogram are equal, how do all the sides compare with each other? COR. 1. In a parallelogram, if one of the angles is a right angle, all are right angles; and if two adjoining sides are equal, the sides are all equal. b.) If the angles of a parallelogram are all equal, what kind of an angle must each be? If, in an oblique parallelogram, each obtuse angle is double each acute, hm° does each angle contain? If each obtuse angle is triple each acute? § 110. c.) How many diagonals (§ 60) can be drawn in a parallelogram? How do the triangles into which a diagonal divides a parallelogram compare with each other? XLIV. A B D 104. C What parts have tADC and the parallelogram ABCD in common? tBAD and ABCD? If a triangle and parallelogram have two sides and the included angle common, what part is the triangle of the parallelogram? Can a triangle and parallelogram have a side and two adjacent angles common? If a triangle and a parallelogram agree in three adjacent parts, what must those parts be? How does the triangle then compare with the parallelogram in extent of surface, or area (§ 16. b)? COR. II. A diagonal divides a parallelogram into two identical triangles. Each of these has two sides and the included angle in common with the parallelogram. |