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m348-assignment1sol

# m348-assignment1sol - Math 348(2006 Assignment#1 Solutions...

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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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m348-assignment1sol - Math 348(2006 Assignment#1 Solutions...

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