. The Public School Euclid and Algebra / authorized for use in the public schools of Ontario on the Department of Education . 22 Euclids elements. PROPOSITION 7. Theorem. Two triangles on the same base and on the same side of it cannothave their contermhwus sides If it is possible let the two As ABC, ABD on the same baseAB, and on the same side of it, have AC equal to AD, and BCequal to BD. Three cases may occur : (1) The vertex of each A may be outside the other A. (2) The vertex of one A may be inside the other A . (3) The vertex of one A may be on a side of the other A .In the first
. The Public School Euclid and Algebra / authorized for use in the public schools of Ontario on the Department of Education . 22 Euclids elements. PROPOSITION 7. Theorem. Two triangles on the same base and on the same side of it cannothave their contermhwus sides If it is possible let the two As ABC, ABD on the same baseAB, and on the same side of it, have AC equal to AD, and BCequal to BD. Three cases may occur : (1) The vertex of each A may be outside the other A. (2) The vertex of one A may be inside the other A . (3) The vertex of one A may be on a side of the other A .In the first case join CD ; and in the second case join CD and produce AC and AD to U and F. Because AC = AD, .-. Z FCD = Z FDC. Prop. 5. But Z FCD is greater than Z BCD. Ax. 9. .-. Z FDC is greater than Z more then is Z BDC greater than Z because BC = BD,.: Z BDC = Z BCD; Prop. 5. that is, Z BDC is greater than and equal to Z BCD,which is third case needs no proof, because BC is not equal toBD. Hence two triangles on the same base and on the sameside of it cannot have their conleiminous sides equal. Euclids elements. 23 PROPOSITION 8. Theorem,If two triangles have tivo sides of the one equal to ttco sides ofthe other, each to ?.ach, and have
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