greater than the angle contained by the sides equal to them, of the other. Let ABC, DEF, be two As, having the sides AB, AC = sides DE, DF, ea. to ea., and base BC > base EF; then / BAC > < EDF. BAC must be either >, =, or < ≤ EDF. and it has been proved / BAC #▲ EDF, EDF. PROP. XXV. THEOR. 26. 1Eu. If two triangles have two angles of the one, equal to two angles of the other, each to each, and one side equal to one side; viz. either the sides adjacent to the equal angles, or the sides opposite to equal angles in each; then shall the other sides be equal, each to each, and also the third angle of the one, equal to the third angle of the other. CASE I. When sides adj. to equals in each be equal. In As ABC, DEF, BCEF, let ABC / DEF, ▲ BCA = / EFD, G A then shall AB =DE, AC = DF, D For if ABDE, then one is > other; but i. e. GB, BC = DE, EF, ea. to ea. GBC= DEF "; base GC base DF, GBCDEF, ▲ GCB=/ DFE; less greater, which is absurd; .. AB is not DE, i e. Then ABDE. JAB, BC = DE, EF, ea. to each, Prop. 3. Prop. 4. Hyp. Hyp. AC = DF, CASE 2. When sides oppo. equals in each then BC= EF, BAC / EDF, D PROP. XXVI. THEOR. 27. 1 Eu. If a straight line falling upon two other straight lines, make the alternate angles equal to one another, these two straight lines shall be parallel. Let str. line EF, which falls upon the str. lines AB, CD, make alt, EFD: then shall AB || CD. AEF alt. For if ABCD, they will, when produced, meet either towards B, D, or towards A, C. Suppose the former, and let them meet in G, then GEF will form a Def. 35. ; which is impossible. ... AB, CD, being produced, do not meet to wards B, D. In like manner it may be proved that they Wherefore if a str. line, &c. Def. 35. D PROP. XXVII. THEOR. 28. 1 Eu. If a straight line falling upon two other straight lines, make the exterior angle equal to the interior and opposite upon the same side of the line; or make the interior angles upon the same side together equal to two right angles; the two straight lines shall be parallel to one another. = Let the str. line EF, which falls upon AB BGH+ GHD 2 rt. Zs; Hyp. Again, GHD= ={ ZAGH, which are alt. LS, AB || CD. Zs, s, BGH+GHD = 2 rt. Prop. 12. and AGH+ BGH = 2 rt. Ax. 1. :: ≤ BGH + ≤ GHD = / AGH + ▲ BGH ; |