Geometry Proofs in Algebra: Properties of Equality

Welcome back to Educator.com.0000

For this lesson, we are going to talk about some properties of equality, and we are going to work on some proofs.0002

Going over some properties first: these are all properties of equality,0013

meaning that they have something to do with them equaling each other, something to do with the word "equal."0017

Now, the first one, the addition property of equality, is when you have, let's say, numbers a, b, and c.0028

If a equaled b, if the number a is the same as b, then if you add c to a, then that is the same thing as adding c to b.0039

So, if a = b, then a + c = b + c; that is the addition property of equality, because you are adding the same number to a and b, since a and b are the same.0051

And the subtraction property of equality: again, you have numbers a, b, and c.0070

a is equal to b; then, a - c is equal to b - c, as long as you subtract the same number.0076

But when you are dealing with subtraction, then it is the subtraction property.0087

But as long as you are subtracting the same number from both sides, then it is still the same.0090

You still have an equation, with equal sides.0096

The multiplication property of equality: again, you have numbers a, b, and c.0101

If a is equal to b, then a times c is equal to b times c; so again, you are multiplying the same number.0106

And the division property: for the numbers a, b, and c, if a is equal to b, then a/c is equal to b/c.0115

Now, here you have to look at this c, because you are dealing with division; so this can also be a/c = b/c.0132

This, even though it is a fraction, also means a divided by c; and when you are dealing with that,0143

since c is now the denominator, we have to keep in mind that c cannot be 0, because we can't have a 0 in the denominator.0150

So, be careful with that.0161

Now, I want to go back over these again; and since the next couple of lessons, we are going to be talking about segments0163

and angles, if I have, let's say, the measure of angle 1, the measure of angle 1 equals the measure of angle 2.0172

So then, if the measure of angle 1 is representing a, and the measure of angle 2 is representing b,0185

then the measure of angle 1, plus the measure of angle 3 (c is a new one) equals...what is b?...0191

the measure of angle 2, plus the measure of angle 3.0200

So, this is also the addition property of equality, but just using angles now.0204

The measure of angle 1, plus the measure of angle 3, equals the measure of angle 2, plus the measure of angle 3.0209

Then, you are adding the same angle measure to both of these sides.0214

The subtraction property is the same thing: if I have, let's say, the measure of angle 1,0220

minus the measure of angle 3, then that is the same thing as the measure of angle 2, minus the measure of angle 3.0230

The multiplication property does the same thing, and the division property would also be the same thing.0240

The reflexive property of equality: this one is when you have one number, a;0252

for every number a, then a equals a--it equals itself; a = a is the reflexive property.0261

You can have a segment AB equaling itself, AB; this is also the reflexive property; measure of angle 1 = measure of angle 1--reflexive property.0271

When you write this, you can write this like "reflexive"...we can write "property"...0286

and for the equality properties, even the ones that we just went over, the addition property,0296

subtraction, multiplication, and division--since they are all properties of equality,0301

you can write "reflexive property of," and then you can write an equals sign next to it, like that: "reflexive property," and then an equals sign.0306

And that equals sign represents the type of property that it is.0314

So, it is the reflexive property of equality.0321

The symmetric property is different than the reflexive property, because you are given two numbers,0326

a and b; you are saying that a equals something else; if a = b, then...and then, you are just going to flip it; and you say b will then equal a.0333

So, if AB = 10, then you can say 10 = AB; and that is the symmetric property.0348

For the symmetric property, you can just write "symmetric property of equality" like that, too.0361

The transitive property of equality: For all numbers a, b, and c, if a = b, and b = c, then a = c.0370

So, let's use angles: if the measure of angle 1 equals the measure of angle 2,0382

and the measure of angle 2 equals the measure of angle 3, then since this and this are the same,0390

the measure of angle 1 equals the measure of angle 3.0401

If this equals that, and that equals something else, then these two will equal each other; and that is the transitive property of equality.0404

This one you can write as "trans. property of equality" for short.0414

A couple more: the substitution property of equality: whenever you replace something in for something that is of the same value,0427

then you are using the substitution property; so if you have numbers a and b, and if a = b,0441

then a may be replaced by b in any equation or expression.0447

If I tell you that x = 4, and x + 5 = 9, then I can take this; since x is equal to 4, I see an x here;0452

so since this and this are the same, I can just replace the 4 in for x...plus 5, equals 9.0475

I am substituting in this for this; and that is the substitution property.0485

For the substitution property, be careful not to just write "sub.," because this can also be the subtraction property.0495

You can just maybe write it like that, or maybe you can write the whole thing out: "substitution property of equality."0504

The distributive property of equality: for all numbers a, b, and c, a times the sum of b and c is equal to ab + ac.0515

Remember: you take this value right here; you multiply it to all the values inside.0527

So, it is going to be a times b, and then plus a times c; and that is the distributive property.0536

You can also go the other way; you can take it from here; you can factor out the a.0548

We have an a in both terms; factor it out; in this term, I have a b, plus...and in this term, I have a c left; that is also considered the distributive property.0553

For the distributive property, you can write it like that; you can write "distributive of equality"; you can write "prop."0565

These are all properties of equality; we are going to be using them pretty often in what is called a proof.0578

And there are a couple of different types of proofs, but the main one is called the two-column proof.0587

And a two-column proof is just a way of organizing your reasoning, and it is deductive reasoning.0594

When you use two-column proofs, you use them to show how to come up with some kind of conclusion.0605

Remember: with deductive reasoning, we talked about having some true statements,0617

and using facts and different definitions and so on to come up with a conclusion.0623

And a two-column proof is just a way of organizing those things.0632

For a two-column proof, you are going to have a given statement, and the given statement is just whatever is given to you,0638

the information that is given; and it can be maybe the values of angles, or the values of the side measures--whatever.0652

Whatever they give you, whatever is given to you, is going to go right here, as given.0662

That is going to be given, and then they are going to give you a "prove" statement, what to prove.0668

Given this information, your conclusion--how will you get to this right here?0677

They are going to give you both statements; and then, on this side, you are going to have a diagram or some kind of drawing,0682

some kind of picture of this proof--some kind of drawing, maybe a diagram; that is going to go right here.0693

And then, right below it, you are going to have something that looks like this.0709

And it is a two-column proof, so you are going to have two columns.0719

On this column, you are going to have statements; on this column, you are going to have reasons.0722

You are going to state something--you are going to state your facts, your different things.0733

And then, on this side, you are going to have reasons for that statement.0739

You can't just say something--you have to have a reason; you have to back it up with something.0744

Why is that statement true? You are going to do numbers 1, 2, 3, 4...and it is going to go on.0748

And then, your reasons: 1, 2, 3, 4...you have to have a reason for every statement you write down.0756

Now, any time you do a two-column proof, the first statement is always going to be your given statement.0763

Whatever is written here, you are also going to write here.0772

You are going to start with your given; and then, your last statement...however many...0776

Now, you don't always have to have 4 or 5; it is usually going to be around 4, 5, or 6, but you can have less; you can have more.0784

It depends on the proof; but your last statement is going to be this statement.0793

Whatever is written here is going to be your last statement.0801

And then, for number 5, you are going to have a reason for that statement.0804

This is what a two-column proof looks like; now, if you are so confused by what a two-column proof is, think of directions.0808

From your house to, let's say, school, or from your house to a friend's house, you have a starting point.0821

You are starting at some place, and you are going to head over to school, or your friend's house, wherever it is.0833

You have directions; if you are to give someone directions to school from your house--or anywhere--0843

the starting point...you have point A to point B; you have steps to get from point A to point B.0850

If you are at home, how are you going to get to point B?0860

You make a right here, make a left here, or whatever it may be; you have directions.0864

There are steps to get there; this is exactly the same thing.0869

The given statement...this is where you start; that is point A--that is your starting point; that is like your house.0874

This statement right here, the "prove" statement, is point B--that is where you have to end up at; that is your destination.0882

You have to go from point A to point B; but again, you can't just snap your fingers and get there.0892

You have steps; you have directions to get there.0901

For each (maybe "make a right turn"; "make a left turn here"), you can't skip any steps, because it has to lead from point A,0905

and then through all of these steps, you are going to end up at point B.0916

And that is what a two-column proof is; they are just saying how you get from here to here.0924

And all you have to do is list out your statements: the starting point, point A, is going to be on line 1, statement 1.0928

Your last statement is going to be right here, your "prove" statement.0937

And you are just going to have reasons for that: why is this statement true? Why is this statement true?0941

Now, when you write your given statement for step 1, your reason is always going to be "Given."0946

That is the reason; this statement is true because it is given--that is given to you.0956

So, step 1 is this part right here, the given statement; and the reason for that is "Given."0964

So, here is an example of a proof: now, here, the statements are just listed, and the reasons are just listed.0975

It is a two-column proof; you can draw a line out like this and draw a line down like this.0983

Or you can just do it like this; as long as you have two columns, a column for statements and a column for reasons, you still have a two-column proof.0992

Again, here is your given statement; you have a few things that they give you; and prove this.1002

So, look at the statements: now, for step 1 (they are not all listed out, so let me write them out here),1011

AC is 21; that is this right here; now, you have to write out all of them.1026

So then, see how only this is written out; so I am going to write in the other ones.1031

AB = 2y, and BC = 3y - 9; those are all of your statements.1037

Now, here, AB is 2y; so I can write that in; so use this diagram to help you get from here, point A, to point B.1050

Write it in: AB is 2y; BC is 3y - 9; and then, AC, the whole thing, is 21.1065

And, given this information, they want you to prove that y equals 6.1080

Step 1: All that I did was to copy down all of the given statements right there.1088

And the reason for that is "Given."1095

Now, the next step: AB + BC = AC...well, that is because, since I have AB, and I have BC, and I have AC,1102

AC, the whole thing, is 21; but since I need to solve for y, I need to look at where my destination is.1119

Where am I trying to get to?--to what y is--my value for y.1127

Well, y, I see here, is from AB, and from BC, not from AC.1131

So, how do I mention these parts, these segments, in relation to the whole thing?1139

This is part of the segment; AB is part, and BC is another part, of this whole segment, AC.1149

If you remember, from chapter 1, we talked about segments, and then their parts.1160

I can say that AB + BC = AC; this part, plus this part, equals the whole thing: AB + BC = AC.1170

And the reason for this step, this statement, AB + BC = AC: if you remember, that is called the Segment Addition Postulate.1186

And you can just write it like this for short: the Segment Addition Postulate.1206

Now, why did I write this down--why is this step here?1214

It is because I know that, in order for me to find the value of y, I have to look at these parts, AB and BC.1218

I can't just look at the whole thing; so when I have to look at the parts, compared to the whole thing,1227

then I have to use the Segment Addition Postulate.1233

And then, what happened here? The next step: 2y + 3y - 9 = 21; so how did I get from this step to this step?1237

What happened here? Well, I know that AB is what?--AB is 2y; BC is this; so, guess what happened right here.1252

You see that...and 21; AC is 21; so, since AB is 2y, just replace AB for 2y, and then replace this for this, and replace this for this.1272

Whenever you do replacing, whenever you replace something for something else, in an equation or expression, that is the substitution property of equality.1292

Now, remember: be careful not to write "sub." because that can mean subtraction; "substitution" is the shortest you can write it.1313

Or you can write the whole thing out.1323

The next step: from here, 5y - 9 = 21--well, how did you get from this step to this step?1327

You did this plus this; you just simplified it, and more specifically, you added; so this would be the addition property (and this is for the "equality").1340

Now, here, number 5: you did 5y, and then you added 9 to both sides; this is the addition property, because you added.1364

And then, from here, how did you get from this step to this step?1385

You divided by 5 on both sides; so this is the division property of equality.1391

And then, since we have this statement, which is the same as this statement right here, we have arrived to our destination, to point B.1406

And once you do that, then you are done; so as long as you start here and you end up here, then you are done.1416

The next example: the measure of angle CDE (there is angle CDE) and the measure of angle EDF are supplementary.1431

Prove that x = 40.1442

So, here I am going to write in...now, for this one, this is x, and this is 3x + 20.1446

Sometimes, they give you the information on the diagram; they might not always give it to you in the given.1466

The given is very important, but you have to look at the diagram, too, because they might label something--1472

an angle, or give you some measure or length, and they might just write it in the diagram.1478

So, that is very, very important to have; if there is no diagram, then draw one in, because that is going to help you.1485

Especially if you are very visual--if you are a visual learner--then you should draw it in and write in whatever is given to you.1491

It will help you with your steps.1502

Number 1: Angle CDE and angle EDF are supplementary.1506

Now, to review over supplementary: supplementary means that two angles add up to 180 degrees.1514

These two angles right here, angle CDE and angle EDF, form a linear pair, meaning that,1529

when you put them together, they form a line; see how there is a line right there--so they are a linear pair.1545

And linear pairs are always supplementary, because a line measures 180 degrees.1555

So, if you have two angles that form a line, then they are supplementary.1563

If you look at supplementary angles, supplementary angles are just any two angles that add up to 180.1571

So, supplementary angles don't always form a linear pair; sometimes they do; sometimes they don't.1576

If you just have two angles that are separated, then they don't form a linear pair, but they can still be supplementary.1582

On to our proof: the reason for #1 is "Given."1591

And then, #2: I want to find x, so if I know that these two angles are supplementary, meaning that they add up to 180,1600

and then I know that these two angles together add up to 180, then I can find x that way.1614

But then, there are steps that I need to take to get there.1621

The next step is going to be that the measure of angle CDE, plus the measure of angle EDF, equals 180.1625

Now, we know that, since it says "supplementary," I can just say, "Well, since they are supplementary, then I add them together, and they equal 180 degrees."1640

And that is because of the definition of supplementary angles.1651

The definition of supplementary angles says that, if two angles are supplementary, then they add up to 180.1662

Any time you go from something supplementary to then making them add up to 180, then that would just be the definition of supplementary angles.1674

Any time you do this step, the reason will be "definition of supplementary angles."1683

The next step: now that I gave these two angles, adding them up to 180, now I have to use x somewhere, because I need to prove that x equals 40.1692

So, this angle right here became x, and then the measure of angle EDF is 3x + 20.1709

So, what happened here? Instead of writing this one, you wrote x; and instead of writing this one, you wrote 3x + 20.1722

Step #3: Since you replaced something, that is the substitution property of equality.1731

Now, #4: This right here, in the last proof--this could be the addition property, because you are adding it.1746

But it could also be the substitution property, because you are just substituting in these two for this value.1759

Let's just write "substitution property of equality."1766

And then, #5: To get from here to here, you subtracted 20 from both sides, so #5 is going to be the subtraction property.1773

And just so that you don't get confused, you can write it out, or you can just write "subtraction property," or "subtract. of equality."1798

In the next step, you divide it by 4 to get x = 40, and that is the division property, because you divided.1808

And then, we know that this is the final step, because that is what that is.1821

OK, another example: The measure of angle AXC and the measure of angle DYF...1831

oh, this is supposed to be written as "equal"; so then they are equal.1848

The measure of angle AXC and this angle are the same; and the measure of angle 1, this one, is equal to the measure of angle 3.1858

And by doing this, this is showing that they are the same.1872

So, if I do this one time, and I do this one time, that means that they are the same.1876

And I have to prove that the measure of angle 2 is equal to the measure of angle 4.1883

For this one, I don't have any statements, so we are going to have to do the statements on our own, and then come up with the reasons as we go along.1892

#1: I am going to write that the measure of angle AXC equals the measure of angle DYF,1899

and that the measure of angle 1 equals the measure of angle 3.1913

OK, and my reason for that is "Given."1925

Now, my next step: since I know that I am trying to prove this and this, that these two are equal,1932

I need to break down this big angle into its parts.1948

I know that angle AXC equals the measure of angle 1 plus the measure of angle 2.1959

So, let me write that out: the measure of angle 1, plus the measure of angle 2, equals the measure of angle AXC.1968

And the reason why I do that is because I need to somehow get that angle 2 in there somewhere.1983

And I know that these equal each other, and I know that the measure of angle 1 and the measure of angle 3 equal each other.1992

How am I going to come up with angle 2?1999

I can say that this one, plus this one, equals this big thing.2006

I am getting it in there somehow: the measure of angle 1, plus the measure of angle 2, equals the measure of angle AXC.2013

Now, I am going to do the same thing for this one, in the same step: the measure of angle 3 plus the measure of angle 4 equals the measure of angle DYF.2018

And the reason for that, if you remember from Chapter 1: this is the Angle Addition Postulate.2040

And my third step: Since all of this equals this, and all of this equals that, look at my first step right here.2060

I know that they equal each other; well, if these equal each other, doesn't that mean that all of its parts equal each other?2077

If this big angle and this big angle equal each other, then angles 1 and 2 together equal angles 3 and 4 together.2085

So, my next step is going to be: The measure of angle 1, plus the measure of angle 2,2095

equals the measure of angle 3, plus the measure of angle 4, because all of this right here equals AXC,2103

and all of this right here equals the measure of angle DYF.2114

And they equal each other; that means that all of this and all of this equal each other; so that is the only step right there.2118

Step 3: I basically just used this right here, and I substituted in the parts for that.2126

So, step 3 is going to be the substitution property; and I can put "equality."2135

Then, for #4: Now, always keep in mind what you have to prove.2146

I have to prove that this one equals this one; so I have to somehow get rid of this and this.2156

Now, look back at step 1; if you look back at step 1, see how the measure of angle 1 equals the measure of angle 3.2167

Well, here is the measure of angle 1, and here is the measure of angle 3.2177

So, since they are the same, I can use the substitution property to replace...2180

maybe measure of angle 3 for this, or measure of angle 1 for that, because they are the same.2186

I am going to just substitute in the measure of angle 1 in place of 3, since they are the same.2194

Do you see that? This is the measure of angle 1, in place of the measure of angle 3, since they are the same, since they equal each other.2209

And my reason for this one is, again, the substitution property; this property is actually used quite often.2218

And then, here, since these are the same, I can just subtract it out.2230

Then, these cancel out; I get that the measure of angle 2 equals the measure of angle 4.2238

My reason is the subtraction property of equality.2247

And I know that I am done, because this stuff is the same as that stuff.2257

Now, I know that this seems really long; but once you get used to it, and once you get more familiar with proofs,2262

they actually become kind of fun, and it is not so long; it is not so bad.2272

It is just that, since we are going over each step, and we are going over each reason, it just seems a lot longer than it is.2276

These next few examples...we are going to just go over the properties that we went over.2289

Name the property of equality that justifies each statement.2304

If 5 = 3x - 4, then 3x - 4 = 5: well, this one right here...remember when we had the property "if a = b, then b = a"?2309

This property is the same property as if I said that if ab = 10, then 10 = ab.2334

And this is the symmetric property of equality, meaning that it is the same on both sides; so you flip it, and it is the same.2347

The next one: If 3 times the difference of x and 3/5 equals 1, then 3x - 5 = 1.2360

So, what happened here--how did you get from this to this?2371

Well, it looks like this was distributed over to everything; this became 3 times x, which is 3x; and 3 times this; 3 times 5/3...2375

this is over 1; you can cross-cancel that out, and that becomes 5.2391

Then, that is how you got that 5; and minus 1; this was the distributive property of equality.2397

You can just also write "distributive...equality."2411

Name the property: Here, this is written: if 2 times the measure of angle 180...2417

no, if 2 times the measure of angle ABC equals 180 (that is how it is supposed to be written), then the measure of angle ABC equals 90.2439

OK, so if 2 times this angle is 180, then if you solve out for the measure of angle ABC, you get 90 degrees.2456

And so, how did you get from this step to that step over there?2467

Well, it looks like you divided the 2; and the measure of angle ABC equals 90; so this one was the division property of equality,2472

because you divided the 2 to get the answer--divided 2 into both sides.2487

Name the property of equality that justifies each statement.2497

For xy, xy = xy; well, this one right here--if something equals itself, this is different than the symmetric property.2501

The symmetric property is when you have something equaling something else.2511

And then, you can reverse it and say that the second thing equals the first thing.2516

In this one, there is no second thing; it is just one thing, and that one thing equals itself, so a = a; apple = apple; xy = xy.2523

Any time you have that, it is the reflexive property; "reflexive," or "reflexive property," and this is used quite often, too, in proofs.2537

If EF = GH, and GH = JK, then EF = JK.2554

Well, if 1 = 2 and 2 = 3, then 1 = 3; this is the transitive property of equality.2563

The next one: if AB + IJ = MX + IJ, then AB = MX.2585

What happened from here to get that? It looks like this happened: this is the subtraction property of equality.2595

The next one: if PQ = 5, and PQ + QR = 7, then 5 + QR = 7.2611

So, this was the equation; there is a value of PQ, and then 5 was replaced for PQ; this is the substitution property of equality.2621

Be careful when you are writing "subtraction" and "substitution."2640

It would probably be best to just write out the whole word.2644

But if you are going to write it like this, then make sure it is obvious what you are writing--subtraction property or substitution property.2648

And that is it for this lesson; we will work on some proofs for the next lesson.2660

So, we will see you next time--thank you for watching Educator.com.2666