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Angles and Trigonometry > Parallel line theorems
A number of theorems involving angles formed by a transversal crossing two parallel lines are handy in Euclidean geometry. Here we establish some of these results in taxicab geometry.
We first note that taxicab distance is invariant under translations, rotations by right angles, reflection about a horizontal or vertical line, or combinations of these transformations (see Schattschneider and Çolakoğlu / Kaya in the references). Since taxicab angles are measured using simple taxicab length, taxicab angles are also invariant under these transformations. This leads to the first lemma.
THEOREM 1: Opposite angles are congruent.
Proof: Using Figure 1 as a guide, note that one angle in a pair of opposite angles can be obtained from the other by a reflection about their point of intersection (a composition of reflections about a horizontal line and a vertical line). Therefore, opposite taxicab angles are congruent.
FIGURE 1: Opposite angles formed by intersecting lines are congruent.
THEOREM 2: Given two parallel lines and a transversal, the alternate interior angles are congruent.
Proof: Using Figure 2, translate alpha along the transversal to become an angle opposite beta. This translated angle has the same measure as alpha. Since opposite angles are congruent, alpha and beta are congruent.
FIGURE 2: Alternate interior angles formed by parallel lines and a transversal are congruent.
 Thompson, Kevin and Tevian Dray. Taxicab Angles and Trigonometry, The Pi Mu Epsilon Journal, Worcester, MA. Vol. 11, No. 2 (Spring 2000), pp. 87-96.
 Schattschneider, Doris J. The Taxicab Group, The American Mathematical Monthly, Vol. 91, No. 7 (Aug-Sep 1984), pp. 423-428.
 Çolakoğlu, Harun Bariş and Rüstem Kaya. Regular Polygons in the Taxicab Plane, Scientific and Professional Journal of the Croatian Society for Geometry and Graphics, Vol. 12 (2008), pp. 27-33.
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