The Public School Euclid and Algebra . ove theorem bear to Prop. 15 ? 3. Show that the bisectors of either pair of vertically opposite angles, in Prop. 15, are in the same straight line. 4. Show that, if ^5 is perpendicular to the straight line CD, which it meets at B, then if A B is produced to E, BE is also perpendicularto CD. 5. From two given points on the same side of a given straight line, show how to draw two straight lines which shall meet at a pointin the given straight line and make equal angles with it. 6. In the figure of Prop. 15, made EB equal to EDy and ^C equal to EA, and join
The Public School Euclid and Algebra . ove theorem bear to Prop. 15 ? 3. Show that the bisectors of either pair of vertically opposite angles, in Prop. 15, are in the same straight line. 4. Show that, if ^5 is perpendicular to the straight line CD, which it meets at B, then if A B is produced to E, BE is also perpendicularto CD. 5. From two given points on the same side of a given straight line, show how to draw two straight lines which shall meet at a pointin the given straight line and make equal angles with it. 6. In the figure of Prop. 15, made EB equal to EDy and ^C equal to EA, and join A D, DB and BC. Then prove the angle .-1 ED equalto the angle CEB, without assuming any proposition after Prop. 5. 7. The side ^C of the triangle ABC is bisected at £, and BE is drawn and produced to F, making EF equal to EB. Show that : fa) FC ^ AB. (h) L. FCE = L BAE. EUCLID S ELEMENTS. 35 PROPOSITION 16. Theorem. If one side of a trianyle he jjroduced, the exterior angle ahidl hegreater than either of the interior opposite Let ABC be a triangle, and let BC be produced to B. It is required to prove _ ACD greater than l BAC, and also greater than _ ABC. Bisect ^1 Cat ^. Prop. \0. Join BE^ and produce it to F, making EF = EB. Prop. 3. Join As ^^EB and CEF, AE = CE, Constr. EB - EF, Constr. and i. AEB = l CEF, Prop. 15. .-. L EAB = L ECF. Prop. 4. But _ ACD is greater than _ ECF; Ax. 9. .-. _ ACD is greater than _ , \i AC be produced to (?,L. BCG is greater than l. ABC. But _ ACD = L BCG; Prop. 15. .. L ACD is greater than L ABC. 36 Euclids elements. PROPOSITIOX 17. Theokem. Any two aiiyles of a triangle are toyetlier leas than two rightangles.
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