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INTRODUCTION 



115. Theorem. Parallelograms on the same base, 

 and between the same parallels, are equal. 



A B C D Since AB=:CD, bolli being eqnal to EF, 



ACziBD (15, or 1(3), and tho triangle AEC 

 is equiangular (105) and equal (102) to BFD ; 

 Y therefore deducting each of them from the 



figure AEFD, the remainder ED is equal to 

 the remainder AF. 



116. Theorem. Parallelograms on equal bases, and 

 between the same parallels, are equal. 



For each is equal to tho parallelogram 

 formed by joining the extremities of the base 

 of the one, and of the side opposite to the base 

 of the other (115). 



117. Theorem. Triangles on equal bases, and be- 

 tween the same parallels, are equal. 



A B j: 1) Take AB and CD equal to the base EF or 



GH, and join BF and DH. Then EB and 

 GD are parallelograms between the same pa- 

 -fl rallels (109), and on equal bases, therefore 

 they are equal (llG), and their halves, the 

 triangles AEF, CGII (114), are also equal (18). 



118. Theorem. In any right angled triangle, the 

 square described on the hypotenuse is equal to the sum of 

 the squares described on the two other sides. 



Draw AB parallel to CD, the side of the 

 square on the hypotenuse, then the parallelo- 

 gram CB is double any triangle on the same 

 base and between the same parallels (114 

 117), as ACD; but ACDziFCG, their angles 

 at C being each equal to ACG increased by a 

 right angle, FC to AC, and GC to DC. Again, 

 GAH is a right line (95), parallel to CF, there- 

 fore the triangle FCG is half of the square CH 

 on the same base, and CH=:C1), since they are the doubles of equal 



