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Archive for the ‘Proofs’ Category

How to Use Definitions in Geometry


Illustration showing angle between curves.

Image via Wikipedia

Time for a post about geometry, which I tutor in addition to algebra and many other subjects.

I especially enjoy helping students learn how to do proofs, which I find is the hardest area of geometry for most kids.

Recently I came up with an analogy to help students understand the special usefulness of definitions in geometric proofs.

The analogy is:  Definitions are like reversible coats.

What?  … you say.

Coats. Reversible coats. As in two for the price of one.

Similarly with definitions:  you get two IF-THEN statements for the price of one when you work with a definition.

Here’s what I mean.

First consider a “standard theorem” in geometry, viewed in the IF-THEN format.

Theorem:  IF two angles are complements of the same angle, THEN they are congruent.

Notice that the converse of this statement doesn’t make much sense:

IF two angles are congruent, THEN they are complements of the same angle. (What other angle? We haven’t even mentioned another angle!)

But when it comes to definitions, you can:

a) First, turn the definition into and IF-THEN statement, and

b) Secondly, you can flip that IF-THEN statement around, and this new statement, called the “converse,” will always be true. You can bank on it!

Example of a definition:  A right angle is an angle that measures 90 degrees.

And here’s one IF-THEN statement that flows out of this definition:

1) IF an angle is a right angle, THEN it measures 90 degrees.

But notice that the converse is also true:

2)  IF an angle measures 90 degrees, THEN it is a right angle.

Let’s try this again, for the definition of perpendicular lines.

Definition: Two lines are perpendicular if they form four right angles.

First IF-THEN statement:

1)  IF two lines are perpendicular, THEN they form four right angles.

Second IF-THEN statement, the converse.

2)  IF two lines form four right angles, THEN the lines are perpendicular.

I am wondering if you are wondering why this is true. Why is it that, for definitions, both the statement and its converse are always true? The reason, I believe, has to do with the nature of a definition. With a definition, we are giving a name to some geometrical object, and stating what we consider to be the defining characteristic of that object.

To take a nonsensical example, suppose that you live in a world that has objects called “Snurfs,” which are measured in units called “Goobles.” Now imagine that some of the Snurfs are special because they have a measure of 100 Goobles. This fact makes these Snurfs so special that you wind up talking about them a lot. And because you talk about them a lot, it is helpful to give them a name. So you do give them a name; you decide to call them “Wombats.” What this means is that anytime a Snurf has a measure of 100 Goobles, you will call it a Wombat. And anytime you see the thing you call a Wombat, you can be sure that it will have a measure of 100 Goobles. For that is just what you have decided the word Wombat will mean. Based on this, you put forth the formal definition:

A Wombat is a Snurf with a measure of 100 Goobles.

Given this definition, notice that you can create two IF-THEN statements:

1)  IF a Snurf is a Wombat, THEN its measure is 100 Goobles.

And you can also state the converse, and it will be true:

2)  IF a Snurf has a measure of 100 Goobles, THEN it is a Wombat.

To me, this is how definitions work. They involve people noticing something they are talking about, and they decide to give it a name so they can talk about it more easily. When they define what the word means, they attach the word to the primary characteristic of this thing, and through this act, the word is born, and along with it, its definition.

Right angle.

Image via Wikipedia

Anyhow, in terms of doing geometry, the important thing to keep in mind is that all definitions can be used reversibly. So, going back to the example of the right angle, here’s what this means.

If, in the course of a proof, you establish that a particular angle is a right angle, you can conclude that the measure of this angle is 90 degrees. Reason:  Definition of a right angle.

And similarly, if in a proof you establish that a particular angle has a measure of 90 degrees, then you can conclude that this angle is a right angle. Reason:  Definition of a right angle.

This reversibility factor is why, when you read through geometric proofs, you will notice that “Definition of … ” is used quite often as a reason for steps. Because they are logically reversible, definitions are TWICE as useful as standard theorems.

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How to See Why the Divisibility Trick for 3 Works


One of my subscribers asked why the trick for divisibility for 3 actually works. [If you missed the post on that trick, go here:]

But the gist of the trick is this:  3 divides evenly into a number if 3 divides evenly into the sum of the digits of the number.

I’ll prove this trick for a three-digit number, but you’ll see why the proof applies to numbers with as many digits as you’d like.

Let’s call our three-digit number cde, where c is the digit in the 100s place, d is the digit in the 10s place, and e is the digit in the 1s place.

We can state the value of our cde number like this:

cde =  (100 x c)  +  (10 x d)  +  e

This shows that the c-part of the number is made up of 100 groups of c. If you think about it, there’s no reason we can’t re-write the value of this digit as 99 groups of c plus 1c, or just:  (99c + c).

In the same way, the d-part of cde is made up of 10 groups of d, which we can re-write as (9d + d). And of course the e-part of the number is just e.

So now we have this:

cde =  (99c +  c)  +  (9d + d)  +  e

Using the rules of jolly old algebra, we can shuffle the terms around a bit to get this:

cde =   99c + 9d + (c + d + e)

Then, adding a set of parentheses for clarity, we get this:

cde =   (99c + 9d) + (cde)

So far, so good. But what’s the point? Well, we’re just getting to that.

Let’s think a bit more about the 99c term. Factoring out a 3, we see that 99c =  3 x 33c. Since 3 is a factor of this expression, 99c must be divisible by 3. Aha, progress, right?

Factoring the d-term, we see that 9d = 3 x 3d. Again, since 3 is a factor of this expression, 9d is  also divisible by 3.

So we now know that both 99c and 9d are divisible by 3.

We can never forget the Divisibility Principle of Sums (DPS), which says:

If a number, x, divides evenly into both a and b, then it divides evenly into their sum, (a + b).

What does that mean here? It means that since 3 divides evenly into both 99c and 9d, it must divide evenly into their sum:   (99c + 9d).

And remember that our number, cde, equals nothing more than:  (99c + 9d) + (cde)

So, we’ve just found out that 3 divides evenly into the quantity in the first parentheses:  99c + 9d

So to find out if 3 divides evenly into the whole number cde, all that’s left is to find out whether or not 3 divides into what remains, the quantity in the second parentheses:  c + d + e

But guess what? c + d + e is the sum of the digits for our number, cde. So this idea right here is the trick for divisibility for 3:  To find out if 3 divides evenly into a number, just add up the number’s digits and see if 3 goes into that sum.  If it does, then 3 does go in; if not, 3 does not go in.

So this proves the divisibility trick for a three-digit number like cde.

To see why the same trick works for numbers with four or more digits, keep in mind that the larger digits can similarly all be broken up as we broke up the digits of c and d. For example, if we have a four-digit number, bcde, then the value of the leading digit b can be viewed, first, as (1,000 x b). And then this can be split apart once more, into 999b + b. That way, this four-digit number fits into the pattern of the trick. And this same kind of split up can be done for any digit whatsoever.

So if you follow the logic of this proof, you now see that the divisibility trick rests on solid logical/mathematical ground.