Honors Geometry Companion Book, Volume 1

2.1.1 Using Inductive Reasoning to Make Conjectures (continued)

This conjecture may appear to be true since it concerns four points and only a quadrilateral is defined by four points. However, the conjecture is that four coplanar points always form a quadrilateral, so if a case can be found where four coplanar points form some figure other than a quadrilateral, then this case will be the counterexample to the conjecture. Notice that in the first figure shown here, all four points are noncollinear. Consider the case where some of the points are collinear. If three of the four points are collinear, then the figure formed is a triangle, which is not a quadrilateral. Therefore, four coplanar points such that three of those points are collinear is a counterexample. Here, the conjecture states that the temperature never exceeds 107° F in Lubbock, Texas, during March, April, and May. So, a counterexample to this conjecture would be any example of a temperature reaching above 107° F during March, April, or May. The table gives the monthly high temperatures in Lubbock for some year. Notice that the high temperature in June was 109° F. However, this is not a counterexample because June is not March, April, or May. So, consider only the high temperatures in March, April, and May. The months of March and April are not a counterexample because the high temperature in those months was not higher than 107° F. But, the high temperature in May was 109° F, which is higher than 107° F. Thus, the month of May is a counterexample.


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