361assignment2 - MTH 361 Assignment 2 Solutions Exercise 3,...

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Unformatted text preview: MTH 361 Assignment 2 Solutions Exercise 3, page 44. Define the following terms: (a) The angles of ABCD are < ) ABC , < ) BCD , < ) CDA , and < ) DAB . The fact that, by the definition of quadrilateral, no three points can lie on the same line ensures that all of these are actually angles (since the rays in question cannot be the same or opposite without three points lying on a single line). (b) Adjacent sides of ABCD are sides that have a point in common. Specifically, the pairs { AB,BC } , { BC,CD } , { CD,DA } , { DA,AB } are adjacent sides. (c) Opposite sides of ABCD are sides with no points in common. Specifically, the pairs { AB,CD } , { BC,AD } are opposite sides. (d) The diagonals of ABCD are the segments AC and BD . (e) A parallelogram is a quadrilateral in which both pairs of opposite sides lie on lines that are parallel to each other. Exercise 8, page 45. Explain why the three given possible definitions of a rectangle are equivalent in Euclidean geometry. Explanation. Before sketching proofs, we need to handle a semantic issue. Despite the fact that the definitions given in the text begin with a capital letter and end with a period, they are not statements (nor even sentences). Hence, it makes no sense to speak of proving them. To remedy this, we convert each to a statement. We replace (i) by the statement, If a quadrilateral has four right angles, then it is a rectangle. Similarly we replace (ii) by, If all four angles of a quadrilateral are congruent, then the quadrilateral is a rectangle, and (iii) by If a quadrilateral is a parallelogram and has at least one right angle, then it is a rectangle. Now we can proceed to show that these statments are equivalent, that is, if any one of them is true then the other two must also hold. A standard way to do this is to prove that (i) implies (ii), that (ii) implies (iii), and that (iii) implies (i). (Make sure you understand why this strategy is valid.) With these preliminaries out of the way, we can proceed with sketching the proofs....
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361assignment2 - MTH 361 Assignment 2 Solutions Exercise 3,...

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