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Vertical angles must necessarily be congruent, however congruent angles do not necessarily have to be vertical angles.

An example of congruent angles which are not vertical angles are the 3 interior angles of an equilateral triangle. These angles do not share the same vertex yet they are congruent.

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Q: What is a counter example for the statement congruent angles are always vertical angles?
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An example that proves that a conjecture or statement is false?

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What is the if-then form of A counter example invalidates a statement?

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Only some statements have both examples and counter examples. A sufficiently clear and unambiguous statement would not have counter examples.


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What does a polygon with all congruent sides is regular mean?

It means that the person writing the statement does not fully understand what "regular" means.A polygon is regular if, and only if, all its sides are congruent andall its angles are congruent.A rhombus, for example, has all its sides congruent but a rhombus is not a regular quadrilateral.It means that the person writing the statement does not fully understand what "regular" means.A polygon is regular if, and only if, all its sides are congruent andall its angles are congruent.A rhombus, for example, has all its sides congruent but a rhombus is not a regular quadrilateral.It means that the person writing the statement does not fully understand what "regular" means.A polygon is regular if, and only if, all its sides are congruent andall its angles are congruent.A rhombus, for example, has all its sides congruent but a rhombus is not a regular quadrilateral.It means that the person writing the statement does not fully understand what "regular" means.A polygon is regular if, and only if, all its sides are congruent andall its angles are congruent.A rhombus, for example, has all its sides congruent but a rhombus is not a regular quadrilateral.


What is counter example?

A counter example is a proof of a negation of a universal statement.A statement of the form "all X are Y" (e.g. all men are mortal), can be disproved by providing a counter example (here: something (someone) which is both a man and immortal).A more mathematical example of the use of a counter example could be to disprove the statement "the product of two prime numbers is odd". This is a claim about all numbers which are the product of two prime numbers (all elements in the set {n in N | n = p*q where p and q are prime numbers}). This set contains infinitely many pair numbers, but a single example (or witness), is enough to disprove the statement. Four is such a number and can serve as a counter example.


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What is a counterexample of All rectangles have two lines of symmetry?

Since the statement does not say that they have exactly two lines of symmetry, I do not believe that there is a counter example.