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Kevin's Corner

<spacer> <spacer> Triangles > Congruent Triangles

In Euclidean geometry a number of conditions will ensure two triangles are congruent. Given two triangles, the triangles are congruent if
  • two angles are congruent and the included side is congruent (angle-side-angle, or ASA), or
  • two sides are congruent and the included angle is congruent (side-angle-side, or SAS), or
  • two angles are congruent and a side opposite one of these angles is congruent (angle-angle-side, or AAS)
In taxicab geometry, a single example will rule out almost all possible congruent triangle conditions. Consider the two triangles shown in Figure 1. The triangle formed by the points (0,0), (2,0), and (2,2) has sides of lengths 2, 2, and 4 and angles of measure 1, 1, and 2 t-radians. The triangle formed by the points (0,0), (2,0), and (1,-1) has sides of length 2 and angles of measure 1, 1, and 2. These two triangles satisfy the ASASA condition but are not congruent. This also eliminates the ASA, SAS, and AAS conditions as well the possibility for a SSA or AAA condition.


FIGURE 1: Triangles satisfying ASASA that are not congruent.

The triangle formed by the points (0,0), (0.5,1.5), and (1.5, 0.5) on the right in Figure 2 has sides of length 2 and angles 1, 1.5, and 1.5 t-radians. Thus, it satisfies the SSS condition with the second triangle in the previous example, also shown at left below. However, the angles of these triangles are not congruent. Hence, the SSS and SSSA conditions fail.


FIGURE 2: Triangles satisfying SSS that are not congruent.

The last remaining condition, SASAS, actually does hold. Its proof relies on the fact that even in this geometry the sum of the angles of a triangle is a constant 4 t-radians (see Basic Triangle Theorem 1). So, two triangles in taxicab geometry are congruent if and only if they satisfy a side-angle-side-angle-side (SASAS) condition.

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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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