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Any two circles in the same plane are congruent. Do you agree or disagree with the conjecture? If you agree, explain how you know. If you disagree, edit the conjecture so it will always be true.

A) Agree, because all circles have the same radius.
B) Disagree, circles can have different radii.
C) Agree, because circles are always congruent.
D) Disagree, circles can have different circumferences.

1 Answer

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Final answer:

The statement that any two circles in the same plane are congruent is false because circles can have different radii and thus different circumferences. Circles are only congruent when they have the same radius.

Step-by-step explanation:

The conjecture that any two circles in the same plane are congruent is incorrect, and therefore, option B is the correct answer. Congruent figures must be identical in size and shape. In the case of circles, for them to be congruent, they must have the exact same radius. However, circles can have differing radii, which causes the size of each circle to differ, therefore, they are not congruent. The circumference of a circle is also directly related to its radius, as it is determined by the formula Circumference = 2πr, where r is the radius. Hence, circles with different radii will also have different circumferences, which supports option D as well. A more accurate conjecture would be: 'Two circles are congruent if they have the same radius.'

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