The elements of Euclid for the use of schools and colleges : comprising the first two books and portions of the eleventh and twelfth books; with notes and exercises . astraight line perpendicular to AB. Take any point D onthe other side of AB, andfrom the centre (7, at thedistance CD, describe thecircle EGF, meeting AB oXF and G. [Postulate 3. Bisect FG at ff, [I. 10. and join CH. The straight line CH drawn from the given point Cshall be perpendicular to the given straight line A B. Join CF, CG. Because FHia equal to HG, [Comfruction. and HC is common to the two triangles FHC, GHC;tlie two sid
The elements of Euclid for the use of schools and colleges : comprising the first two books and portions of the eleventh and twelfth books; with notes and exercises . astraight line perpendicular to AB. Take any point D onthe other side of AB, andfrom the centre (7, at thedistance CD, describe thecircle EGF, meeting AB oXF and G. [Postulate 3. Bisect FG at ff, [I. 10. and join CH. The straight line CH drawn from the given point Cshall be perpendicular to the given straight line A B. Join CF, CG. Because FHia equal to HG, [Comfruction. and HC is common to the two triangles FHC, GHC;tlie two sides FH^ HG are equal to the two sides GH, HC,each to each ; and the base CF is equal to the base CG; [Definition the angle CHF is equal to the angle CHG ; [I. they are adjacent angles. But when a straight line, standing on another straight line,makes the adjacent angles equal to one another, each of tlieangles is called a right angle, and the straight line whichstands on the other is called a perijendicular to it. [Def. 10. Wherefore a perpendicular CH has been draicn tothe given straight line AB from, the given point C with-out it. Q£.f. 2. ^18 EUCLIDS ELEMENTS, PROPOSITION 13. THEOREM, The angles which one straight line makes irith anotherstraight line on one side of it, either are two right angles,or are together equal to two right angles. Let the straight line AB make with the straight lineCD, on one side of it, the angles CBA, ABD: these eitherare two right angles, or are together equal to two rightangles. / For if the angle CBA is equal to the angle ABD, eachof them is a right angle. [Definition 10. But if not, from the point B draw BE at right angles toCD; [I. 11. therefore the angles CBE, EBD are two right angles.[Df/.10. Now the angle CBE is equal to the two angles CBA, ABE; to each of these equals add the angle EBD ; therefore the angles CBE, EBD are equal to the three angles CBA, ABE, EBD. [Axiom 2. Again, the angle DBA is equal to the two angles D
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