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Once you've created a Venn diagram, how can you use it to verify a conjecture?

To verify a statement means to prove that it is true--a fact, not just a good guess. Verifying a conjecture involves making sure that the conjecture is true, or at least seems to be. It's important to realize, though, that Venn diagrams only help with using informal proofs to verify conjectures. A proof cannot be formal unless you show the statement is true for every example possible. A Venn diagram like that would be too large to see all at once!

Let's use the Venn diagram you just created to verify some conjectures.

polygon venn diagram

Conjecture 1

Conjecture 2

Conjecture 3

Conjecture: All quadrilaterals have at least one obtuse interior angle.

Question

In the Venn diagram above, there is a circle that holds all the quadrilaterals. Are there any counterexamples to this conjecture? In other words, is there a quadrilateral that does not have at least one obtuse angle?

There is a rectangle in the quadrilateral section of the diagram. A rectangle has four right angles, which means that there is at least one quadrilateral that does not have at least one obtuse angle. This counterexample disproves the conjecture above.

Conjecture: All convex shapes are quadrilaterals.


Question

Use the Venn diagram to find a counterexample to this conjecture. When you think you have an answer, click the show me button to compare the counterexample you found to ours.

A triangle is convex, but it is not a quadrilateral. A triangle is just one possible counterexamples.

So far we have used Venn diagrams to show that some conjectures cannot be verified. Now, what about this one?

Conjecture: All convex polygons have interior angles less than 180°.

Question

First, look at all the examples of convex polygons that our Venn diagram has. Are there any counterexamples?

No, that area of the Venn diagram does not include any counterexamples.

Question

Now check all the polygons that do not appear inside the convex polygon region (because they are concave, not convex). Do any of the shapes in these areas have angles that are greater than 180°?

Each concave polygon has an interior angle that is greater than 180°.
If your Venn diagram is full of examples that support the conjecture, then you can say that you have informally verified the conjecture.

Question

Using the Venn diagram, what evidence can help verify the conjecture: Concave quadrilaterals have at least one angle greater than or equal to 90°.

Of the examples gathered so far, all the concave quadrilaterals in the Venn diagram have at least one angle greater than 90°.