Plane and solid geometry . Ex. 142. Construct a triangle ABC^ given two sides and an altitudeto one of the given sides. Abbreviate thus : given a, 6, and hb. Ex. 143. Construct a triangle ABC^ given a side, an adjacent angle,and the altitude to the side opposite the given angle. Abbreviate thus:given a, -B, and ht. Ex. 144. Construct an isosceles triangle, given an arm and the me-dian to it. Ex. 145. Construct an isosceles triangle, given an arm and the anglewhich the median to it makes with it. Ex. 146. Construct a triangle, given two sides and the angle which amedian to one side makes : (a)
Plane and solid geometry . Ex. 142. Construct a triangle ABC^ given two sides and an altitudeto one of the given sides. Abbreviate thus : given a, 6, and hb. Ex. 143. Construct a triangle ABC^ given a side, an adjacent angle,and the altitude to the side opposite the given angle. Abbreviate thus:given a, -B, and ht. Ex. 144. Construct an isosceles triangle, given an arm and the me-dian to it. Ex. 145. Construct an isosceles triangle, given an arm and the anglewhich the median to it makes with it. Ex. 146. Construct a triangle, given two sides and the angle which amedian to one side makes : (a) with that side ; {h) with the other side. Ex. 147. Construct a triangle, given a side, an adjacent angle, andits bisector. 62 PLANE GEOMETRY \ Proposition XIV. Theorem 153. If one side of a triangle is prolonged, the exteriorangle formed is greater than either of the remote in-terior Griven A ABC with AC prolonged to D, making exterior Z prove Z. DCB > Z. ABC ov Z. CAB. Argument 1. Let E be the mid-point of BCj draw AE, and pro-long it to Fj making EF= AE. Draw CF. 2. In A ABE and EFQ, BE = EC, 3. AE = EF. 4. Z BE A = Z CEF. 5, .\AABE = AEFC. 6. \Zb = Z fce. Reasons 1. A str. line may be drawn from any one point toany other. § 54, 15. 2. By cons. E is the mid- point of BG, 3. By cons. 4. If two str. lines inter- sect, the vertical A areequal. § 77. 6. 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. Homol. parts of equal fig-ures are equal. § 110. BOOK I 53 Argument 7. Z. DCB > Z FCE, 8. •. Z DCB -> 9. Likewise, if ^C is pro- longed to G, ZacG>/.CAB. 10. But Z DCB = Z ACG. 11. A Z DCB >/CAB. 12. .•• Zdcb> Zabg or Z (7^5. Reasons 7. The whole > any of its parts. § 54, 12. 8. Substituting Z i? for its equal Z i^C^. 9. By bisecting line
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