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By Daniel Tell Sigley; W T Stratton

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1, L 1 = L 2 and L3 = L4. 1 The figure formed-by two parallel lines gives rise to many interesting and useful relations, especially when the figure is extended to include one or more lines parallel to or intersecting them. In Euclidean geometry, all properties of sets of parallel lines depend upon the fundamental assumption that there is one and only one line through an exterior point parallel to a given line (Postulate 8). This chapter will be devoted to the study of the properties of configurations that include sets of parallel and/or perpendicular lines.

9. Congruent Triangles. A given geometric figure is assumed to he rigid and consequently not affected by motions of rotation, translation, or a combination of rotations and translations. Accordingly a given figure can be moved about freely (Postulate 9) without disturbing the interrelation of its parts. This concept is not strictly in line with practical analogues of geometric figures, such as wire frames and other flimsy structures, but for purposes of comparing one figure with another such an assumption is convenient.

Draw an equilateral triangle ABC two inches on a side. Divide the base into four equal parts and designate the successive segments by AF, FG, GE, and EB. Draw CF and CEo Prove that the triangle CFE is isosceles. CONGRUENCE OF TRIANGLES 37 11. 6 inches, respectively. Draw a perpendicular from the vertex to the longest side of the triangle and measure this altitude. 12. To find the distance AB across a lake, a surveyor selected a point C on the shore from which A and B were visible and made the following measurements: LACB = 72°, CA = 450 ft, and CB = 400 ft.

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