Geometry Inductive Reasoning

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This next lesson is on inductive reasoning.0001

For inductive reasoning, we deal with what is called conjectures; a conjecture is an educated guess.0008

When you look at several different situations, or maybe previous experiences, to come up with a final conclusion, then that would be inductive reasoning.0017

When you have repeated observations, or you look at patterns, those things would be considered inductive reasoning.0036

Basically, you are just looking at past experiences--anything that will lead you to some sort of conclusion is inductive reasoning.0042

Looking at patterns: if I have 4, 8, 16, 32, and I need to use inductive reasoning0057

to find the next several terms in the sequence, well, I can just see how these numbers came about,0067

and then I can just apply the same rule to find the next few numbers.0073

So, for this, 4, 8, 16, 32...how did you get from 4 to 8, 8 to 16, and 16 to 32?0079

Well, it looks like this was multiplied by 2; that means I would have to multiply by 2 to get my next answer.0091

So, if I multiply this number by 2, then I am going to get 64; if I multiply this by 2, then I am going to get 128; and so on.0103

Now, for this next one, I have to see...well, here is a triangle, a square, triangle, triangle, square, square...what will be my next shape?0121

Well, it went from 1 triangle, 1 square, to 2 triangles, 2 squares; so my conjecture will be that it will be 3 triangles, and then 3 squares.0134

Again, we are looking at patterns, or looking at some kind of repeated behavior, to come up with a conclusion (or what will happen next).0155

Now, a few more problems: if we make a conjecture about this, if AB = CD and CD = EF, what can I conclude?0169

Well, if AB is equal to CD, if I draw AB, here is AB; and here is CD;0184

so, if I see that this and this are the same, CD = EF, so this equals this; isn't it true that, if this equals this and this equals this...0198

doesn't that mean that AB will equal EF?--so that will be my conjecture: AB = EF.0217

Make a conjecture, given points A, B, and C: let me just draw out a coordinate plane.0229

A is (-1,0); it is right here; B is (0,2); it is right there; C is (1,4), which is right there.0256

If I look at this, A, B, and C line up; so my conjecture would be that points A, B, and C are collinear, because they are on the same line.0272

Counter-examples: just because you come up with a conjecture, you come up with some kind of conclusion,0299

based on what you see in your observations, based on the patterns, and so on, doesn't mean that it is going to be true.0306

Just because something happens 3, 4, 5, or 6 times in a row doesn't mean that it is going to happen again the next time.0313

Conjectures are not always true; and to prove that it is not always true, you have to provide a counter-example.0322

And a counter-example is the opposite of what you are trying to prove.0331

If you are trying to prove that something is true--let's say you saw something a few times,0336

and so you conclude--you make a conjecture--that the next time, it is going to happen again;0343

you can't prove that it is going to happen again just by showing that it happened.0349

You cannot prove something just by giving an example, because it might not happen the following time.0354

You might not be able to find a counter-example in order to prove that that is not true.0364

A counter-example is the opposite of what you are trying to prove.0373

Let's say, for example, that the first five cars you see today are black; does that mean that all cars are black?0376

That would be a conjecture; the conjecture would be that, since you saw five cars that are black...0385

my conjecture would be that the next car that I see will be black; and that might not be true.0392

In order to prove that whatever you concluded, your conjecture, is not true, you are going to provide a counter-example.0403

A counter-example would be to show an example of it not being true.0413

Now, let's go over a few examples of this: the first one: Any three points will form a triangle.0418

If I have three points like this, I know that it is going to form a triangle.0427

Is this conjecture true? It could be true, but just because I gave an example of it being true does not make this conjecture true,0440

because I know that three points will not always form a triangle.0451

And so, what I can do to prove that this is not true--to prove that it is false: I can give an example of when this is not true.0458

And that would be a counter-example: so three points that do not form a triangle...there are three points; they don't form a triangle.0466

This is my counter-example: by giving an example of when this is not true, I am proving my conjecture false.0481

So, this conjecture...sometimes it could be true, but it is not always true.0493

By showing an example of when it is not true, a counter-example--that is when you are proving the conjecture false.0502

The square of any number is greater than the original number: well, that could be true, but it is not always true.0512

To show that it is not always true, I need to provide a counter-example.0519

Let's say I have some numbers: let's say 2--if I square it...I am saying the square of any number,0523

so if I take a number, and I square it, then it is going to be greater than this original number.0531

If I square this, then it is going to be 4; well, this is greater than this number, the original number.0536

What if I have 0? If I square it, what do I get? 0.0546

If I have, let's say, 1/2, and I square this, I get 1/4.0553

Well, is 1/4 greater than the original number, 1/2? No, 1/4 is smaller than 1/2.0564

So, for this example, this is true; but this one is not true; this is false, and this is false, for this being greater than the original number.0571

0 is not greater than 0, and 1/4 is not greater than 1/2.0582

Here is a counter-example, and here is a counter-example, because these two examples show that,0594

if you square the number, then the answer is not going to be greater than the original number.0602

Counter-example, counter-example: by showing the counter-examples, I am proving that this conjecture is false.0610

Find the pattern and the next two terms in the sequence: 15 to 12, 9 to 6...the pattern here that I see is subtracting 3.0623

So, from here to here, I subtract 3; subtract 3; subtract 3; if you subtract 3 again, then you get 3; you get 0, -3, and so on.0637

This one: 1 to 2, 2 to 6, 6 to 24, and so on--this one seems a little tricky, but you just have to look at it.0655

Here, if I...let's see...1 to 2: I can either add 1; I can multiply by 2; here I can add 4, or I can multiply by 3;0667

here I can add something bigger...for this one, it doesn't seem like that would be the pattern;0683

so here it is multiplied by 4; here, multiply by 5; so then, for the next number, I can multiply by 6.0691

If I multiply this by 6, then I will get...let's see: 120 times 6 is 0, 12, 720.0706

And the next one...you can just multiply by 7.0721

This one right here: I have a square; then I have this shape in there; and then I have another one.0728

The next pattern will be...so then, here is the next step; that is up to there.0736

And then, my next one will be to draw a square within that square, like that.0750

And the next one would be to do the same thing; and you are going to draw another square inside.0758

OK, so you are given a statement, and you need to come up with a conjecture.0769

The given statement is that angle 1 and angle 2 are adjacent.0776

"Adjacent" means that the angles share a side and a vertex.0781

So, if I draw angles 1 and 2, they are adjacent; then what can I conclude?0787

Well, I conclude that angles 1 and 2 (since I know that "adjacent" means that they are next to each other) share a side;0803

so, angles 1 and 2 share a side and a vertex, because they are adjacent.0817

The next one: the given statement: line m (here is line m, and this is a line) is an angle bisector of angle ABC.0842

Here is an angle, ABC; and line m is an angle bisector--"bisector" means that it cuts in half.0855

So, this line cuts this angle ABC in half; that means that these two parts right here are the same--they are congruent.0867

If I label this angle 1 and this angle 2, what can I conclude?0881

If this line is an angle bisector, then these two parts right here, angle 1 and angle 2, I can say are congruent.0887

Or I could say that the measure of angle 1 equals the measure of angle 2.0895

Or I can say that angle 1 is congruent to angle 2.0902

That would be my conjector, since line m is an angle bisector of angle ABC.0909

The next example: Decide if each conjecture is true or false; if true, then explain why; and if it is false, then we have to give a counter-example.0918

Given that WX = XY, the conjecture is that W, X, and Y are collinear points.0932

The conjecture is saying that if I have WX...here is point W, point X, and point Y...W, X, and Y are collinear.0947

WX = XY; the conjecture is that these are collinear; is that always true?0960

Can you think of an example of when it is not true?0968

Well, what if you have WX, XY...WX is equal to XY; this still applies here, but it doesn't prove that the conjecture is true.0971

This is a counter-example, because this is an example of when my conjecture is false.0997

This one I know...this conjecture is false.1005

The next one: Given that x is an integer, the conjecture is that -x is negative.1012

So, if x is an integer (some integers are 2, 0, and -2; so these are x), the conjecture is that -x is negative.1019

If I make this negative, -2 is -x--is it negative?--these would be -x.1033

-2: this is true; this is an example of the conjecture; what about 0?1045

If I make it negative x, then that will be -0, which is just 0; and that is not a negative, so this is false.1055

And then here, -2: if I make it -x, then it is -(-2); well, that is a positive 2.1067

So, does that show that -x is negative? No, because this is positive 2.1076

So, in this case, this one is false; if x is an integer (these numbers right here), and you make those integers negative,1083

then the answer is going to be negative: in this case, this works.1097

If you make this number negative, it is not a negative; if you take the negative of this number, it becomes positive.1103

So, -x is positive in this case; so this one is also false, and then here is my counter-example, right here.1112

This one and this one are both counter-examples, because they show that this is not true; it is false.1122

Well, that is it for this lesson; thank you for watching Educator.com--see you next time!1135