Plane and solid geometry . DE DFAB AG 4 DE DF.•. = • DH DK \ 5. .. HK 11 EF, 6. .-. A DEF ^ A DHK, It remains to prove A DHK = A ^BC. 7. DE EF DH HK 8. -D . DE EF . DEiiut — = —; — = AB BC DH EF BC 9. /. HK=BC. 10 I^ow DH= AB and DK = AC. 11. .-. A DHK=A ABC, 12. But A DEFr^ A DHK. 13. .. A DEF ^ A ^BC. EF _DFBc AC Reasons 1. §54,14 2. §54,15. 3. By hyp. 4 §309. 5. §415. 6. §423. 7. §424,2. 8. By hyp. 9. §391. 10. Arg. 1. 11. §116. 12. Arg. 6. 13. §309. Ex. 673. Tf the sides of two triangles are 0, 12, lo, and 6, 8, larespectively, are the tiiangles similar ? Explain. BOOK III 181
Plane and solid geometry . DE DFAB AG 4 DE DF.•. = • DH DK \ 5. .. HK 11 EF, 6. .-. A DEF ^ A DHK, It remains to prove A DHK = A ^BC. 7. DE EF DH HK 8. -D . DE EF . DEiiut — = —; — = AB BC DH EF BC 9. /. HK=BC. 10 I^ow DH= AB and DK = AC. 11. .-. A DHK=A ABC, 12. But A DEFr^ A DHK. 13. .. A DEF ^ A ^BC. EF _DFBc AC Reasons 1. §54,14 2. §54,15. 3. By hyp. 4 §309. 5. §415. 6. §423. 7. §424,2. 8. By hyp. 9. §391. 10. Arg. 1. 11. §116. 12. Arg. 6. 13. §309. Ex. 673. Tf the sides of two triangles are 0, 12, lo, and 6, 8, larespectively, are the tiiangles similar ? Explain. BOOK III 181 Ex. 674. Construct a triangle that shall have a given perimeter andshall be similar to a given triangle. Ex. 675. Construct a trapezoid, given the two bases and the twodiagonals. Hint. How do the diagonals of a trapezoid divide each other ? Proposition XYI. Theorem 428. If two triangles have an angle of one equal toan angle of tlie other, and the including sides propor-tional, the triangles are
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