18 INTRODUCTION. 



y In the triangles ADC, DBF, let AC=I>F, 

 BC=iEF, and L ACBzzDFE. Now supposing 

 a triangle equal to DEF to be constructed on 

 AC, the side equal to FE must coincide in po- 

 sition with CB, because Z. ACBziDFE, and 

 also in magnitude, for tliey are equal, therefore the point B will be 

 an angular point of the supposed triangle ; and since the base of both 

 triangles must be a right line, it must be the same line AB (81), and 

 the supposed triangle will coincide every where with ABC ; there- 

 fore ABCinDEF, and tlie angles at A and B are equal to the angles 

 at D and E. 



87. Theorem. If two sides of a triangle are equal, 

 the angles opposite to them are equal. 



In the sides AB and AC produced, take at pleasure 

 AD=AE, and join BE, CD; then since ADzzAE, 

 and ACizAB, and the angle at A is common to the 

 triangles ADC, AEB, those triangles are equal (86), 

 and L ACDziABE, L ADC=:AEB, and CDzrBE ; 

 but BDzzCE (16), therefore L BCDziCBE (86), 

 and Z. ACD— BCDziABE— CBE (16), or L ACB=:ABC. 



88. Theorem. If two angles of a triangle are equal, 

 the sides opposite to them are equal. 



^ Let Z ABC^zBCD ; then AC=AB. If it bo de- 



nied, take, in the greater AC, CD equal to the less 

 AB; then, since Z. ABC=:DCB, ABzzDC, and 



B^ ^C BC is common, the triangle ABCz=DCB (86), the 



whole to a part, which is impossible. 



89. Theorem. If two triangles have their bases equal, 

 and their sides respectively equal, their angles are also 

 respectively equal. 



C F If a triangle be supposed to be constructed 



^^ /j on AB, the base of ABC, equal to DEF, the 

 vertex of the triangle must coincide with C, and 

 » T» ^> the whole triangle with ABC. For if it be de- 

 nied, let G be the vertex of the triangle so con- 

 structed ; join CG ; then since AC=: AG, Z. ACG=:AGC (87), and 



