Parallels and Euclidean Geometry Lines l and m which are coplanar but do not meet are said to be parallel; we denote this by writing l m. Likewise, segments or rays are parallel if they are subsets of parallel lines. Given any two lines m and n, any third line l that meets both m and n in distinct points is called a transversal. This configuration makes a total of 8 angles with l, four with m and another four with n. Let l and m meet at A and let l and n meet at B. Any one of the angles at A is said to be alternate with respect to any angle at B that lies on the opposite side of l. Either of the two angles at A whose sides include B are called interior angles, as are either of the two angles at B whose sides contain A; the other four angles are exterior angles of the configuration. Finally, any two of these angles not at the same vertex and on the same side of l which are not both interior angles or both exterior angles are said to be corresponding angles. A central story in the history of geometry is the reaction to the manner in which Euclid developed the concept of parallelism in his Elements. A study of Book I shows that the first 28 Propositions were developed in absolute geometry; this was Proposition 28:
Theorem If two coplanar lines are cut by a transversal so that some pair of alternate interior angles are congruent, then the two lines are parallel. // Proposition i.29 is the converse of this theorem, but it is not a theorem of absolute geometry: to prove it, Euclid made use of a new axiom, his last and Fifth Postulate: If two coplanar lines are cut by a transversal so that some pair of interior angles on the same side of the transversal have total measure less than 180, then the two lines meet on that side of the transversal. This postulate is effectively equivalent to Proposition i.29: If two coplanar parallel lines are cut by a transversal, then both pairs of alternate interior angles are congruent. Following Euclid, we will also add a new axiom to absolute geometry which is logically equivalent to, but simpler than, the Fifth Postulate. It was formulated by the Greek geometer Proclus in the 5th century CE, but is more often attributed to John Playfair (the Scottish geometry professor who in 1795 published a classroom edition of the Elements with this variant replacing the Fifth Postulate). [P-1] For any line l and point P not on l, there exists in the same plane as l and P a unique line passing through P and parallel to l.
Theorem If two coplanar parallel lines are cut by a transversal, then both pairs of interior angles on the same side of the transversal have total measure equal to 180. // From this we deduce a number of useful transversal properties. (See Figure 4.7, p. 215, for an explanation of why they have these names.) Corollary [The C Property] If two coplanar lines and only if some pair of interior angles on the same side of the transversal are supplementary. // Corollary [The F Property] If two coplanar lines and only if some pair of corresponding angles are congruent. // Corollary [The Z Property] If two coplanar lines and only if some pair of alternate interior angles are congruent. // Theorem Parallelism is an equivalence relation on the set of all lines. //
Theorem [The Exterior Angle Theorem] The measure of any exterior angle of a triangle equals the sum of the measures of the two opposite interior angles. // Corollary [Euclidean Triangle Angle Sums] The angle sum of any triangle is 180. // Theorem [The Midpoint Connector Theorem] The segment joining the midpoints of two sides of a triangle is parallel to the third side and has half the length. // A convex quadrilateral is called a parallelogram if both pairs of opposite sides are parallel, a rhombus if, in addition, some pair of adjacent sides are congruent, and a square if also one of its interior angles is right. Theorem Either diagonal of a parallelogram divides it into two congruent triangles. // Corollary Both pairs of opposite sides of a parallelogram are congruent. // Corollary Any pair of adjacent angles in a parallelogram are supplementary. // Corollary If a convex quadrilateral has both pairs of opposite sides congruent, it is a parallelogram. //
Corollary If a convex quadrilateral has a pair of opposite sides congruent and parallel, it is a parallelogram. // Corollary The diagonals of a convex quadrilateral bisect each other if and only if it is a parallelogram. // Theorem A parallelogram is a rhombus if and only if its diagonals are perpendicular. // Theorem A parallelogram is a rectangle if and only if its diagonals are congruent. // Theorem A parallelogram is a square if and only if its diagonals are perpendicular and congruent. // A trapezoid is a convex quadrilateral having one pair of opposite sides, called the bases, parallel to each other. The other pair of opposite sides are called its legs. (Note that all parallelograms are trapezoids.) The median of a trapezoid is the segment joining the midpoints of its legs. Theorem [The Midpoint Connector Theorem for Trapezoids] A line parallel to the bases of a trapezoid that bisects one leg bisects the other; conversely, the median of a trapezoid is parallel to its bases. Further, the length of the median is the average of the lengths of the bases. //