of the Twelfth Axiom of Euclid. 



265 



angles comprised by them and a third side are respectively equal, 

 the other two angles are equal. 



(c) In a quadrangle, if two adjacent angles are equal and the 

 two opposite sides comprising them are unequal, the other two 

 angles are unequal. 



(d) In a quadrangle, if two adjacent angles are equal and the 

 other two angles are unequal, the opposite sides comprising the 

 adjacent equal angles are unequal. 



Obs. — When the opposite sides comprising two adjacent equal 

 angles are unequal, the longest of those sides will comprise the 

 smallest of the two unequal angles. 



First Principal Proposition. 



In a quadrangle, if two opposite sides are equal and two adja- 

 cent angles comprised by them and a third side are equal and 

 obtuse, the other two angles are equal and acute. 



Jn the quadrangle ABCD Fig. 1. 



(fig. 1), let the sides AB, CD 

 be equal, and the angles A B D, 

 B D C be equal and obtuse. At 

 the point B, on the side AB, 

 elevate within the quadrangle a 

 perpendicular B N ; sufficiently 

 prolonged it will meet A C at C or at a point E between A and 

 C ; and ABC (or A B E) being a right angle, B A C must be 

 acute ; A C D is equal to B A C ; therefore the angles B AC, ACD 

 are equal and acute. Or it will 

 meet the side C D (fig. 2) at a 

 point E between C and D. In that 

 case, prolong B E from E until its 

 prolongation E F is equal to B E, 

 and from F draw a perpendicular 

 FCtoCD. IfitfallatC, NCD 

 being a right angle (complement 

 of ECF), ACD must be acute; 

 therefore B A C, A C D are equal 

 point G (fig. 3) in the prolon 



Fig. 2. 



BAC is equal to ACD; 

 and acute. If it fall at a 



gation of D C, AC sufficiently 

 prolonged from C will meet 

 GFor AFat Horl. In the 

 first case ACD equal of the 

 acute angle GCH is acute. In 

 the second case, A B I being a 

 right angle, B A I or BAC 

 must be acute, and thus, as in 

 the preceding cases, the angles 

 BAC, A C D are demonstrated 



Fig. 3. 



equal and acute. If the per- 



