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Kevin's Corner

<spacer> <spacer> Triangles > Triangle Basics

One fundamental feature of triangles in Euclidean geometry is the sum of the angles is always π radians. A very similar result exists in taxicab geometry. (For information about t-radians and angles in taxicab geometry, see the Angles page.)

THEOREM 1: The sum of the angles of a triangle is 4 t-radians.

Proof: Given the triangle in Figure 1, we can translate the angle gamma from Q to R and by the congruence of alternate interior angles (see Parallel Line Theorem 2) conclude the sum of the angles of the triangle is 4 t-radians.


FIGURE 1: The sum of the angles of a taxicab triangle is always 4 t-radians.

Some other very familiar features of Euclidean triangles fail to carry over, however. For example, the triangle shown in Figure 2 is an equilateral triangle (all sides have the same taxicab length). But, the angles of this triangle are 2 t-radians, 1 t-radian, and 1 t-radian. Therefore, equilateral taxicab triangles are not necessarily equiangular.


FIGURE 2: An equliateral taxicab triangle that is not equiangular.

In addition, the base angles of an isoceles taxicab triangle need not be congruent as illustrated in Figure 3. The base angles have measure 2 t-radians (a right angle) and 4/3 t-radians.


FIGURE 3: An isoceles taxicab triangle whose base angles are not congruent.

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References
[1] 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.
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