Find a counterexample to show that the converse of each conditional is false.

1. If x=-5, then x^2=25.

2. If two angles are adjacent, then they share a common vertex.

Please help!:) thank you!

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- October 1st 2011, 07:59 AMChristyh101Proofs/ counterexamples
**Find a counterexample to show that the converse of each conditional is false.**

1. If x=-5, then x^2=25.

2. If two angles are adjacent, then they share a common vertex.

Please help!:) thank you! - October 1st 2011, 08:11 AMPlatoRe: Proofs/ counterexamples
- October 1st 2011, 09:54 AMSorobanRe: Proofs/ counterexamples
Hello, Christyh101!

Do you know what*adjacent angles*are?

Then you can surely come up with a counterexample . . .

Quote:

Find a counterexample to show that the converse of the conditional is false.

2. If two angles are adjacent, then they share a common vertex.

The converse is:

. . If two angles share a common vertex, then they are adjacent.

*Adjacent angles*are angles that share a common side.

The converse is false.

We can have two angles with a common vertex,

. . which do**not**share a common side.

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- October 1st 2011, 10:29 AMDevenoRe: Proofs/ counterexamples
in general (and this is worth thinking about now) when you have a statement like "if A, then B" it means that A is a stronger statement than B (and is true for fewer things).

for example, x = -5 tells us everything we need to know about x. only one x will make this statement true, namely, -5.

by contrast, x^2 = 25 only tells us "something" about x, that its square is 25. this happens to be true for two values of x, one is -5, another is 5.

similarly, A and B are adjacent angles, tells us two pieces of information: they share a vertex, and a common side.

this is clearly stronger than: A and B share a common vertex, which is only one piece of information (weaker).

one thing you should have clear in your mind, always: if A is stronger than B, we may not be able to derive A from B.

IF we can, then A and B are "the same strength", that is, they are equivalent statements (true for the same things).

so:

A implies B

A is stronger, true for fewer (or the same number of) things, more restrictive

B is weaker, true for more (or the same number of) things, more inclusive

you will come upon this "shape of thought" in many areas: logical thinking, mathematics (of many kinds), action and consequence. learn it well.