OF SPACE. 23 



107. Problem. Through a given point to draw a right 

 line parallel to a given right line. 



From A draw, at pleasure, AB, meeting BC in B, \ d 



ami make /.BADz:ABC (101), then AD||CB "^^--^^^^ 

 (104> c B - 



108. Theorem. The angles of any triangle, tdkcn 

 together, arc equal to two right angles. 



Produce A B to C, and draw BD parallel to AE. ^ jj 



Then ^EBDrzAEB (105), and ZDBCziEAB; 

 therefore the external angle EBC is equal to the 

 sum of the internal opposite angles, AEB, EAB, and A B C 



adding ABE, the sum of all three is equal to ABE+EBC, or to two 

 right angles (94). 



109. Theorem. Right lines, joining the extremities of 

 equal and parallel right Hues, are also equal and parallel. 



Let AB and CD be equal, and parallel. Then a B 



A C will be equal and parallel to BD. For, joining 

 BC, /.ABCzzBCD (106), and the triangles ABC, 

 DCB, are equal (86), and AC=DB ; also Z.ACB 

 ziDBC, therefore AC 1 j BD (104). 



110. Definition. A figure, of which the opposite 

 sides are parallel, is called a parallelogram. 



111. Definition. A straight line, joining the oppo- 

 site angles of a parallelogram, is called its diagonal. 



112. Definition. A parallelogram, of which the 

 angles are right angles, is a rectangle. 



113. Definition. An equilateral rectangle is a 

 square. 



114. Theorem. The diagonal of a parallelogram 

 divides it into two equal triangles, and its opposite sides 

 are equal. 



For ABC is equiangular with DCB (105), and A B 



BC is common, therefore they are equal (102), \ 

 and AB=CD, and AC=:BD. ^' 



C D 



