Plane and solid geometry . Fig. 1. Fig. 2. Ex. 67. Extend Ex. QQ> to the case in which the equal segmentsare laid off on the arms prolonged through the vertex (Fig. 2). 30 PLANE GEOMETRY pROPOsiTiox IV. Theoremm. The base angles of an isosceles triangle are Given isosceles A ABC, with AB and BC its equal prove /. A = Z. C, Argument Reasons 1. Let BD bisect Z ABC. 1. Every Z has but one bi-sector. § 53. 0 In A ABD and DBC,AB = BC, 2. By hyp. 3. BD = BD. 3. By iden. 4. Z1 = Z2. 4. By cons. 5. .-. aabd = Adbc. 5. Two A are equal if twosides and the includedZ of one are equal r
Plane and solid geometry . Fig. 1. Fig. 2. Ex. 67. Extend Ex. QQ> to the case in which the equal segmentsare laid off on the arms prolonged through the vertex (Fig. 2). 30 PLANE GEOMETRY pROPOsiTiox IV. Theoremm. The base angles of an isosceles triangle are Given isosceles A ABC, with AB and BC its equal prove /. A = Z. C, Argument Reasons 1. Let BD bisect Z ABC. 1. Every Z has but one bi-sector. § 53. 0 In A ABD and DBC,AB = BC, 2. By hyp. 3. BD = BD. 3. By iden. 4. Z1 = Z2. 4. By cons. 5. .-. aabd = Adbc. 5. Two A are equal if twosides and the includedZ of one are equal re-spectively to two sidesand the included Z ofthe other. § 107. 6 ,\ZA=Za 6. Homol. parts of equal fig- ures are equal. § 110. 112. Cor. I. The bisector of the angle at the vertex ofan isosceles triangle is perpendicular to the base andbisects it, 113. Cor. n. An equilateral triangle is also equi-angular. Ex. 68. The bisectors of the base angles of an isosceles triangle are euual. BOOK I 31 114. Historical Note. Exercise 69 is known as the pons asi-novum, or bridge of asses, since it has proved difficult to many beginnersin geometry. This propositionand the proof here suggestedare due to Euclid, a greatmathematician who wrote thefirst systematic text
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