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ROYAL SOCIETY OF CANADA 



Theorem 10. — Two triangles are congruent if a side in each and 

 the adjacent angles are congruent. 



Let BC E B'C, < ABC ^^ y^ 



E < A' B' C, < A C B E 

 > A'C'B'. 



IfABbenotE A'B',take 

 B'A"EBA. Then[Thm. 9] 

 the triangles A B C, A" B' C 

 are congruent, and therefore 



< ACBE< A'C'B'. But 



< A CBE< A'C'B'. There- B c JB' c' 

 fore [III., (5)] < B' C A" ^< W C A', which is contrary to iH., (4). 

 Hence A B E A' B', and therefore [Thm. 9] the triangles are congruent. 



After shewing that if 

 < A O B E < A' O' B' 

 and<BOCE<B'0'C', 

 then<'AOCE< A'O'C; 

 and also that the angles 

 at the base of an isosceles 

 triangle are congruent^ 



Â' 



we are able to shew that if the three sides of one triangle are respectively 

 congruent to the three sides of another, the triangles are congruent : 



Let<C'B'A"E<CBA, 

 and B' A "E BA. 



Then triangles ABC, 

 A" B' C are congruent. 

 Therefore < B' A' A" E 



< B'A"A', and < C'A'A" 

 E < C A" A'. Therefore 



< B' A'C'E<B' A"C'E 



< B A C ; and the triangles 

 A B C, A' B' C are con- 

 gruent [Thm. 9]. 



From the above we see 

 that if A', B', C be three 

 non co-straight points, a point A" exists such that A" b' E A' B' and 

 A"C'EA'C'. 



We may also reach the general proposition that if i\. B C . . . , 

 A' B' C . . . be congruent figures, and P any point whatever, then there 

 exists a point P', such that the figures A B C . . . P, A' B' C • . . P' are 



