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Unformatted text preview: Math 348 (2006): Assignment #1: Solutions 1. Consider the correspondence of vertices A A , B C , C B . Under this cor- respondence, we have that B = C (by hypothesis) and also C = B . Finally, the sides contained between these corresponding pairs of vertices are also congruent: BC = CB because they are the same segment. Thus by the ASA criterion for con- gruence, 4 ABC = 4 ACB . Consequently, all the corresponding sides are congruent, so AB = AC and the triangle is isosceles. 2. Let 2 ABCD be a rectangle. We know that it has four right angles, and also that its diagonals bisect each other and are congruent. For the first direction, suppose that the rectangle is a square. This means all 4 sides are congruent. Let M be the point of intersection of the two diagonals AC and BD . Consider 4 DMA and 4 AMB . We know that DM = AM = MB since the diagonals are congruent and are bisected at M . Also DA = AB since it is a square. Therefore by SSS we have that 4 DMA = 4 AMB ....
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