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Proving Angle Relationships

  • Supplement Theorem: If two angles form a linear pair, then they are supplementary angles
  • Congruence of Angles: Congruence of angles is reflexive, symmetric, and transitive
  • Angle Theorems:
    • Angles supplementary to the same angle or to congruent angles are congruent
    • Angles complementary to the same angle or to congruent angles are congruent
    • All right angles are congruent
    • Vertical angles are congruent
    • Perpendicular lines intersect to form four right angles

Proving Angle Relationships

If ∠ABC and ∠CBD form a linear pair, and m∠ ABC = m∠CBD
Find m∠ABC.
  • ∠ ABC and ∠ CBD form a linear pair (Given)
  • m ∠ABC + m ∠CBD = 180o (Definition of linear pair)
  • m ∠ABC = m ∠CBD (Given)
  • m ∠ABC + m ∠ABC = 180o (substitution property of equality)
m ∠ABC = 90o (Division property of equality)
Complete the statement with always, sometimes, or never.
Two angles that are supplementary to the same angle are ____ congruent.
Always.
Complete the statement with always, sometimes, or never.
Two supplementary angles are _____ right angles.
Sometimes.
Complete the statement with always, sometimes, or never.
Vertical angles are _____ congruent.
Always.
Complete the statement with always, sometimes, or never.
Two lines intersect, they _____ form four right angles.
Sometimes.
Complete the statement with always, sometimes, or never.
Congruent angles are ______ right angles.
Sometimes.

Line AB and line CD intersect at point O, m∠AOC = 4x + 3, m∠COB = 5x + 6, find m∠AOC and m∠COB.
  • ∠ AOC and ∠ BOC are supplementary angles
  • m ∠AOC + m ∠COB = 180
  • 4x + 3 + 5x + 6 = 180
  • 9x = 171
  • x = 19
  • m ∠AOC = 4x + 3 = 4*19 + 3 = 79
m∠BOC = 5x + 6 = 5*19 + 6 = 101.
∠1 and ∠2 are both complementary to ∠3. m ∠1 = 2x + 3, m ∠2 = 3x + 4, find x.
  • ∠1 is congruent to ∠2.
  • m∠1 = m∠2
  • 2x + 3 = 3x + 4
x = − 1.
∠1 and ∠2 are vertical angles, ∠2 and ∠3 are completary angles, m ∠1 = 25o, find m ∠3.
  • ∠1 and ∠2 congruent
  • m ∠2 = m ∠1 = 25o
  • ∠2 and ∠3 are complementary angles
  • m ∠2 + m ∠3 = 90o
m ∠3 = 90o − m ∠2 = 90o − 25o = 65o.
Justify the statement with a property of congruence of angles.
If 1 ≅ 2, then 2 ≅ 1.
Symmetric

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

Proving Angle Relationships

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

Transcription: Proving Angle Relationships

Welcome back to Educator.com.0000

This next lesson is on proving angle relationships.0002

Let's go over the supplement theorem; this lesson actually uses a lot of theorems.0007

And remember, from the last lesson: a theorem is a statement that has to be proved.0012

It is not like a postulate, where we can just assume them to be true; the theorems have to be proved in order for us to use them.0020

And usually, your book will prove it for you; and then, after that, you can use it whenever you need to.0029

The first one (this is the supplement theorem) is "If two angles form a linear pair, then they are supplementary angles."0035

Now, supplementary angles are two angles that add up to 180 degrees.0045

A linear pair would be two angles, a pair of angles, that form a line--"linear" means line.0059

Two angles, a pair of angles, that form a line are supplementary angles.0070

That means that if the two angles form a line, then they will add up to 180.0078

If angle 1 and angle 2 form a linear pair (here is angle 1, and here is angle 2)--because you put them together, they form a line--0085

and the measure of angle 1 is 85 (this is 85), then find the measure of angle 2.0095

Then, this is what we are trying to find.0102

Well, we know that, since these two angles form a linear pair, they are supplementary; that means that they add up to 180.0105

And then, if the measure of angle 1 is 85, well, this plus this one is 180, so I can say 85 degrees, plus the measure of angle 2, equals 180.0115

This is the measure of angle 1 that is given; and then, the measure of angle 2...together, they add up to 180.0130

If I subtract 85, what will I get here? The measure of angle 2 is 95 degrees.0138

That is how I am going to find the measure of angle 2--using supplementary angles and the supplement theorem.0150

OK, congruence of angles is reflexive, symmetric, and transitive.0159

Reflexive: remember, reflexive is when a = a; that is the reflexive property.0167

The symmetric property is just "if a = b, then b = a."0173

And then, the transitive property is, "if a = b, and b = c, then a = c."0178

So, if you need to review those properties, then just go back a few lessons before when we talked about each of those in more detail.0185

But congruence of angles: when we talk about angle congruence, then these properties also apply.0197

This is a proof, because this is a theorem, so we have to prove that.0208

And we are just going to prove one of these; so in your book, they will actually show you all of the proofs for each of these,0216

but for this lesson, we are just going to just prove one of them.0224

Let's see, angle 1 is congruent to angle 2: we know that these two are congruent;0229

angle 2 is congruent to angle 3, so prove that angle 1 is congruent to angle 3.0236

Which property does that sound like? It sounds like the transitive property.0243

We are just saying that we are going to prove that the transitive property can be applied to congruent angles.0251

#1: For statements, and then reasons on this side...statement #1: Angle 1 is congruent to angle 2, and angle 2 is congruent to angle 3.0260

The reason for that, we know, is "Given"; the first one is always given.0285

#2: Since we are trying to prove that the transitive property works for congruent angles, I have to first,0291

because I know that the transitive property will work for angles that are equal, change this to measure of angle 1 = measure of angle 2,0304

and measure of angle 2 = measure of angle 3; and in that case, all I did was change the congruence to the "equals."0317

That is the definition of congruent angles.0327

And then, the third one: If the measure of angle 1 equals the measure of angle 2, and the measure of angle 20335

equals the measure of angle 3, then I know that I can apply the transitive property to this one.0341

So, this is "the measure of angle 1 equals the measure of angle 3," and this would be the transitive property.0347

And I know that that is the transitive property.0359

I can use the transitive property, because it is the "equal."0363

And then, #4: I am trying to prove that angle 1 is congruent to angle 3; angle 1 is congruent to angle 3, and all I did there was to go from equal to congruent.0367

So, it is the definition of congruent angles.0381

So then, here, this is the proof to show that if you have this given to you, you can use the transitive property.0388

You can prove that the transitive property works for the congruence of angles, and that is just all it is.0401

Now, you can use these three properties on the congruence of angles.0406

Angle theorems: #1: Angles supplementary to the same angle or to congruent angles are congruent.0416

Now, these theorems in your book are probably labeled Theorem 2-Something; make sure you don't use that name.0426

Don't call it by whatever your book calls it; the only time that you can use the same name is when there is an actual name for the theorem.0435

But don't call it by what the book labels it, if it is 2-Something, because all of the books will be different.0444

And if there is no name for it (like number 1--there is no name for it; it is just the theorem itself), then you would have to write out the whole theorem.0453

But there is a way for you to abbreviate it: Angles supplementary to the same angle, or to congruent angles, are congruent:0462

so then, you can just say, "Angles supplementary to same angle or to congruent angles are congruent."0470

You can just write it like that; any time you have "angles," you just write the angle sign; if you have "congruent," then write the congruent sign.0482

And then, you can just shorten "supplementary."0491

Now, what is this saying? "Angles supplementary to the same angle or to congruent angles are congruent."0496

Well, if I have an angle, say angle 1; and let's say this angle is supplementary to angle A (this is angle A);0504

angle 1 and angle A are supplementary angles--that means that, if you add them up0526

(you add the measure of this angle and add the measure of that angle), you get 180.0532

Now, let's say I have another angle, angle 2: now, this angle is also supplementary to angle A.0537

Then, they are saying that these two angles have to be congruent.0549

If two angles are supplementary to the same angle, then they have to be congruent.0554

And if this is 100 degrees (the measure of that is 100), then this has to be 80.0560

There is only one angle supplement to this, and there is only one angle supplement to that.0569

So, if these two are supplementary, and these two are supplementary, the only angle measure that this can be is 100.0578

I can't find another angle measure that will be supplementary to this angle that is different than 100.0588

As long as they are both supplementary to the same angle, then they have to be congruent.0598

You can't have two angles supplementary with different measures.0602

That is what this is saying: if angles are supplementary to the same angle (both of these are supplementary to the same angle, A), then they are congruent.0611

And the same thing here: Angles complementary to the same angle or to congruent angles are congruent.0626

If I have...here is angle 1, and that is complementary to angle A; and then I have angle 2;0631

if this is, let's say, 70, and angle 1 and angle A are supplementary, then this has to be 20 degrees, because "complementary" is 90.0649

So, if the measure of angle 1 is 70, and the measure of angle A is 20,0670

and let's say that the measure of angle 2 is also complementary to angle A;0675

then this has to also be 70, so they are going to be congruent; angle 1 is congruent to angle 2.0679

More theorems: All right angles are congruent.0694

If I have a right angle, the measure of that angle is 90; if I have a right angle like this, guess what that is--90.0701

If I have one like this, that is still 90; so all right angles are congruent, because they all have the same measure.0718

Vertical angles are congruent: this one is actually going to be used a lot.0730

Remember: vertical angles are angles like this one and this one.0737

These are vertical angles, and these are vertical angles; so that means that these two, the angles that are opposite each other, are going to be congruent.0746

And then, these two angles that are opposite each other are also congruent.0756

Vertical angles are always congruent; they are not congruent to each other--make sure that you don't get it confused with this one and this one.0762

It is always going to be the opposite, so this and this.0772

Perpendicular lines intersect to form four right angles.0778

If I have perpendicular lines, this is 90; this one right here (remember, linear pairs are always supplementary)--0782

if this is 90, then this one has to also be 90, because they form a line.0799

And then, this angle and this angle form a line; that is a linear pair, so this has to be 90; and the same thing here--this has to be 90.0805

They all become right angles; as long as the lines are perpendicular, you have four right angles.0815

Complete each statement with "always," "sometimes," or "never."0827

Two angles that are complementary to the same angle are ___ congruent.0831

Two angles that are complementary to the same angle--I have two angles, and they are complementary to the same angle, A.0839

This is complementary to A, and angle 2 is complementary to A; then they have to always be congruent.0856

Vertical angles are [always/sometimes/never] complementary.0869

Vertical angles, we know, are like this; so, vertical angles are complementary...0873

well, we know that vertical angles are congruent; that means that,0884

if they are to be complementary, then that means they have to add up to 90.0887

But then, they have to be the same measure; so if they are going to add up to be 90, then each of them has to be 45.0894

In that case, then the vertical angles would be complementary, because they add up to be 90 degrees.0904

But look at these angles right here: these angles are not complementary, so what would this angle be?0911

This would be 135; how do I know that?--because this is a linear pair, and they are supplementary.0921

This one and this one...135 + 45 has to add up to 180, because it is a line; they form a linear pair.0930

This angle and this angle are congruent, because they are vertical; vertical angles are always congruent.0939

But they are not complementary; so in this case they can be complementary, but in this case they are not.0948

So, here, this would be my counter-example--an example that shows that something is not true,0955

an example that shows where this statement is not going to be true.0963

Then, I know that it could be complementary, and it might not be complementary.0969

So, this will be sometimes.0976

Two right angles are ___ supplementary; two right angles are [always/sometimes/never] supplementary.0982

If I have a right angle, that is 90; if I have another right angle, that is 90; 90 and 90 always make 180.0992

So, this will be always; two right angles are always supplementary.1004

OK, let's do a few examples: Two angles that are supplementary [always/sometimes/never] form a linear pair.1015

Let me think of a counter-example: try to think of two angles that are supplementary that do not form a linear pair.1028

Well, how about if I have an angle like this; let's say this is 100 degrees, and then I have another angle like this that is 80 degrees.1039

They are supplementary, right? Yes, they are, because they add up to 180.1057

Do they form a linear pair? No, so this is my counter-example.1061

Can they form a linear pair, though?--yes, because, if I have a linear pair, here is 100 and here is 80.1075

So, my answer will be sometimes.1088

Two angles that form a linear pair are [always/sometimes/never] supplementary.1097

Two angles that form a linear pair--that means that we have to have a linear pair--are [always/sometimes/never] supplementary.1102

It is always, because a linear pair will always be supplementary.1116

I can make this 120; then this will be 60; if I make this 100, it is going to be 80.1123

No matter what, they have to be supplementary, because they form a linear pair.1129

Find the measure of each numbered angle: the measure of angle 1 is 2x - 5; the measure of angle 2 is x - 4.1137

What do I know about the measure of angle 1 and the measure of angle 2?1147

They form a linear pair, so then, they are supplementary; they add up to 180.1150

Make sure you don't make them equal to each other, because they are not.1159

This is obviously an obtuse angle, and this is an acute angle.1164

And they don't look like they are the same; besides that, you just know that you can't always assume1169

that two angles that are adjacent are going to be congruent, or that they are going to be equal or have the same measure.1175

Keep in mind that two angles that form a linear pair--always remember that a linear pair's angles are going to add up to 180.1184

So, I am going to take the measure of angle 1, 2x - 5; I am going to add it to the measure of angle 2: + x - 4.1192

And then, I am going to make them equal to 180.1201

See how this was the measure of angle 1 and the measure of angle 2, like that.1205

And then, I just solve it out; so then, 2x + x is 3x; -5 - 4 is -9; that equals 180.1218

I add the 9; it becomes 189; and then I divide by 3, and that is going to give me 63.1229

OK, so then, that is my x; and then, what is it asking for?1242

They are asking for the measure of each numbered angle; so then, be careful--when you solve for x, you don't leave it like that.1250

You have to plug it back in; if they were asking for x, then yes, that would be the answer.1259

But in this problem, they are not; they are asking for the measure of the numbered angle.1266

Then, you have to plug it back in, because they want you to find the measure of angle 1 and the measure of angle 2.1271

So then, for the measure of angle 1, 2x - 5...I have to do 2(63) - 5; and that is going to be 126 - 5; that is 121, so it is 121.1277

The measure of angle 2 is 63 - 4, which is 59.1302

And then, make sure; you can double-check your answer by adding them up, because if we add them up, then they should add up to 180, and this does.1309

So, here is the measure of angle 1, and here is the measure of angle 2.1320

Measure of angle 3 and measure of angle 4--their relationship: they are vertical.1329

We know that vertical angles are congruent; so in this case, since they are vertical, I can make them equal to each other.1336

This is not supplementary, and don't assume that they are complementary.1347

They could be supplementary, and they could be complementary, but we don't know that they are.1352

What we do know for sure is that they are congruent; they are the same.1358

So, I can just make them equal to each other.1362

So, the measure of angle 3 is going to equal the measure of angle 4.1366

The measure of angle 3 is 228 - 3x = x; that is the measure of angle 3, and this is the measure of angle 4.1374

If I add 3x to that side, I am going to get 228 = 4x.1386

And then, I divide the 4, I am going to get 57.1391

And then, again, they want you to find the measure of the numbered angle, not x.1402

So, we take the x, which is 57, and we are going to plug it back in.1414

The measure of angle 3 equals 228 - 3(57); and then, 228 minus...this is going to be...171.1419

And then, subtract it, and you are going to get 57; and the measure of angle 4 is going to be 57.1450

Is that right? Well, what do we know?1477

Now, unlike this problem right here, where we can just add them up and then see if they are supplementary,1480

because we know that they are supplementary, we can't do that to this, because they are not supplementary; they are congruent.1484

We have to check our answer; we have to just see that they have the same measure.1492

And they do; so then, that would be the answer for the measure of angle 3 and the measure of angle 4.1496

OK, find the measure of each numbered angle; angle 1 and angle A are complementary; angle 2 and angle A are complementary.1505

Since we know that this is complementary to this and this is complementary to this, what do we know?--1519

that the measure of angle 1 equals the measure of angle 2.1525

If two angles are complementary to the same angle, angle A, then they are congruent.1530

Angle 1 is complementary to angle A; angle 2 is complementary to angle A; that means that these have to be congruent.1538

I can just make these angles equal to each other, and then I just substitute in 2x + 25.1546

And then, this is going to be x = 20.1559

And then again, I have to look back and see: OK, they want me to find the measure of each number angle,1567

so I have to find the measure of angle 1 and the measure of angle 2.1572

I have to plug x back in: so the measure of angle 1 equals 3(20) + 5.1575

Let's see: here I am going to have 65, and then, even though I know that they are the same value,1584

that the measure of angle 1 is going to equal the measure of angle 2, I still want to plug in x and see if I am going to get the same number.1600

2 times 20, plus 25--this is 40, and that is 65.1610

So then, I do have the same measure, so that shows me that it is right.1617

The next example: here, we have a proof; we are going to write a two-column proof.1628

Our given statements are that the measure of angle ABC is equal to the measure of angle DEF, or DFE; this should actually be DEF;1635

and the measure of angle 1 equals the measure of angle 4.1656

We are going to prove that the measure of angle 2 is equal to the measure of angle 3.1661

So, remember: my column here is going to be statements; my column here is going to be reasons.1665

Remember: my first step is going to be the given statements: so the measure of angle ABC equals the measure of angle DEF.1680

And then, the reason for that is going to be "Given."1694

Number 2: Let's look at this really quickly, or think about this.1700

I have that this big angle is equal to this big angle; they have the same measures.1709

And this part, this angle, is equal to the measure of this angle.1717

And then, I want to prove that the other part of it is going to be equal to this part.1727

Remember: when we have a bigger angle, and we want to break it down into its parts1733

(because that is what we are doing: we are dealing with the big angle's parts, and the same thing on this side),1738

then we want to use the Angle Addition Postulate, because that is what breaks it down from its whole to its parts.1746

The measure of angle ABC equals the measure of angle 1, plus the measure of angle 2.1758

The measure of angle 1 plus the measure of angle 2 equals this big thing.1770

And the same thing for the other one: the measure of angle DEF equals the measure of angle 4, plus the measure of angle 3.1773

And the reason is the Angle Addition Postulate.1797

Now, from here, since I know that the measure of angle 1 is equal to the measure of angle 4,1810

and I want to prove that these two are equal to each other, I can just make this whole thing equal to this whole thing.1815

So, I am going to use step 1, and I am going to replace all of this with its parts.1825

This is the formula; this is equal to this--that is what is going on with this one: the measure of angle ABC is equal to the measure of angle DEF.1835

I am going to replace the measure of angle ABC with its parts, which is this,1846

and then replace all of that with its parts there.1854

And the reason is going to be the substitution property--and I can write "equality."1862

And then, again, the measure of angle 1 and the measure of angle 4 are equal to each other.1876

So, I can just replace one of them with the other; so I just replace this with the measure of angle 4,1884

because it is given; and I should actually write that, too, for here, because this is another given statement:1898

the measure of angle 1 equals the measure of angle 4.1906

Since they are equal to each other, I can just replace one for the other.1911

And whenever I do that, that is also the substitution property (I am running out of room).1917

Then, here, since these two are the same, I can just subtract it out.1925

If I subtract it out, then, the measure of angle 2 equals the measure of angle 3.1934

And right here, this is the subtraction property of equality.1941

And is that my "prove" statement? Yes, it is, so I am done.1956

So, remember to always keep in mind...because, from here, you can go in a lot of different directions;1962

you can make a left; you can make a right; you can take any direction; it depends on your destination--where are you trying to get to?1970

You are trying to get to this right here; if you are trying to get to this statement right here, you have to lead it from this step to this step.1979

Maybe after you write your given statement, or before you even start, just look at it and think,1991

"OK, well, if I want to get from here to here, what steps do I have to take--what do I have to do?"1997

And once your last statement is the same as this right here, then you are done.2005

That is it for this lesson; thank you for watching Educator.com--we will see you next time.2013

I. Tools of Geometry
  Coordinate Plane 16:41
   Intro 0:00 
   The Coordinate System 0:12 
    Coordinate Plane: X-axis and Y-axis 0:15 
    Quadrants 1:02 
    Origin 2:00 
    Ordered Pair 2:17 
   Coordinate Plane 2:59 
    Example: Writing Coordinates 3:01 
   Coordinate Plane, cont. 4:15 
    Example: Graphing & Coordinate Plane 4:17 
    Collinear 5:58 
   Extra Example 1: Writing Coordinates & Quadrants 7:34 
   Extra Example 2: Quadrants 8:52 
   Extra Example 3: Graphing & Coordinate Plane 10:58 
   Extra Example 4: Collinear 12:50 
  Points, Lines and Planes 17:17
   Intro 0:00 
   Points 0:07 
    Definition and Example of Points 0:09 
   Lines 0:50 
    Definition and Example of Lines 0:51 
   Planes 2:59 
    Definition and Example of Planes 3:00 
   Drawing and Labeling 4:40 
    Example 1: Drawing and Labeling 4:41 
    Example 2: Drawing and Labeling 5:54 
    Example 3: Drawing and Labeling 6:41 
    Example 4: Drawing and Labeling 8:23 
   Extra Example 1: Points, Lines and Planes 10:19 
   Extra Example 2: Naming Figures 11:16 
   Extra Example 3: Points, Lines and Planes 12:35 
   Extra Example 4: Draw and Label 14:44 
  Measuring Segments 31:31
   Intro 0:00 
   Segments 0:06 
    Examples of Segments 0:08 
   Ruler Postulate 1:30 
    Ruler Postulate 1:31 
   Segment Addition Postulate 5:02 
    Example and Definition of Segment Addition Postulate 5:03 
   Segment Addition Postulate 8:01 
    Example 1: Segment Addition Postulate 8:04 
    Example 2: Segment Addition Postulate 11:15 
   Pythagorean Theorem 12:36 
    Definition of Pythagorean Theorem 12:37 
   Pythagorean Theorem, cont. 15:49 
    Example: Pythagorean Theorem 15:50 
   Distance Formula 16:48 
    Example and Definition of Distance Formula 16:49 
   Extra Example 1: Find Each Measure 20:32 
   Extra Example 2: Find the Missing Measure 22:11 
   Extra Example 3: Find the Distance Between the Two Points 25:36 
   Extra Example 4: Pythagorean Theorem 29:33 
  Midpoints and Segment Congruence 42:26
   Intro 0:00 
   Definition of Midpoint 0:07 
    Midpoint 0:10 
   Midpoint Formulas 1:30 
    Midpoint Formula: On a Number Line 1:45 
    Midpoint Formula: In a Coordinate Plane 2:50 
   Midpoint 4:40 
    Example: Midpoint on a Number Line 4:43 
   Midpoint 6:05 
    Example: Midpoint in a Coordinate Plane 6:06 
   Midpoint 8:28 
    Example 1 8:30 
    Example 2 13:01 
   Segment Bisector 15:14 
    Definition and Example of Segment Bisector 15:15 
   Proofs 17:27 
    Theorem 17:53 
    Proof 18:21 
   Midpoint Theorem 19:37 
    Example: Proof & Midpoint Theorem 19:38 
   Extra Example 1: Midpoint on a Number Line 23:44 
   Extra Example 2: Drawing Diagrams 26:25 
   Extra Example 3: Midpoint 29:14 
   Extra Example 4: Segment Bisector 33:21 
  Angles 42:34
   Intro 0:00 
   Angles 0:05 
    Angle 0:07 
    Ray 0:23 
    Opposite Rays 2:09 
   Angles 3:22 
    Example: Naming Angle 3:23 
   Angles 6:39 
    Interior, Exterior, Angle 6:40 
    Measure and Degrees 7:38 
   Protractor Postulate 8:37 
    Example: Protractor Postulate 8:38 
   Angle Addition Postulate 11:41 
    Example: Angle addition Postulate 11:42 
   Classifying Angles 14:10 
    Acute Angle 14:16 
    Right Angles 14:30 
    Obtuse Angle 14:41 
   Angle Bisector 15:02 
    Example: Angle Bisector 15:04 
   Angle Relationships 16:43 
    Adjacent Angles 16:47 
    Vertical Angles 17:49 
    Linear Pair 19:40 
   Angle Relationships 20:31 
    Right Angles 20:32 
    Supplementary Angles 21:15 
    Complementary Angles 21:33 
   Extra Example 1: Angles 24:08 
   Extra Example 2: Angles 29:06 
   Extra Example 3: Angles 32:05 
   Extra Example 4 Angles 35:44 
II. Reasoning & Proof
  Inductive Reasoning 19:00
   Intro 0:00 
   Inductive Reasoning 0:05 
    Conjecture 0:06 
    Inductive Reasoning 0:15 
   Examples 0:55 
    Example: Sequence 0:56 
    More Example: Sequence 2:00 
   Using Inductive Reasoning 2:50 
    Example: Conjecture 2:51 
    More Example: Conjecture 3:48 
   Counterexamples 4:56 
    Counterexample 4:58 
   Extra Example 1: Conjecture 6:59 
   Extra Example 2: Sequence and Pattern 10:20 
   Extra Example 3: Inductive Reasoning 12:46 
   Extra Example 4: Conjecture and Counterexample 15:17 
  Conditional Statements 42:47
   Intro 0:00 
   If Then Statements 0:05 
    If Then Statements 0:06 
   Other Forms 2:29 
    Example: Without Then 2:40 
    Example: Using When 3:03 
    Example: Hypothesis 3:24 
   Identify the Hypothesis and Conclusion 3:52 
    Example 1: Hypothesis and Conclusion 3:58 
    Example 2: Hypothesis and Conclusion 4:31 
    Example 3: Hypothesis and Conclusion 5:38 
   Write in If Then Form 6:16 
    Example 1: Write in If Then Form 6:23 
    Example 2: Write in If Then Form 6:57 
    Example 3: Write in If Then Form 7:39 
   Other Statements 8:40 
    Other Statements 8:41 
   Converse Statements 9:18 
    Converse Statements 9:20 
   Converses and Counterexamples 11:04 
    Converses and Counterexamples 11:05 
    Example 1: Converses and Counterexamples 12:02 
    Example 2: Converses and Counterexamples 15:10 
    Example 3: Converses and Counterexamples 17:08 
   Inverse Statement 19:58 
    Definition and Example 19:59 
   Inverse Statement 21:46 
    Example 1: Inverse and Counterexample 21:47 
    Example 2: Inverse and Counterexample 23:34 
   Contrapositive Statement 25:20 
    Definition and Example 25:21 
   Contrapositive Statement 26:58 
    Example: Contrapositive Statement 27:00 
   Summary 29:03 
    Summary of Lesson 29:04 
   Extra Example 1: Hypothesis and Conclusion 32:20 
   Extra Example 2: If-Then Form 33:23 
   Extra Example 3: Converse, Inverse, and Contrapositive 34:54 
   Extra Example 4: Converse, Inverse, and Contrapositive 37:56 
  Point, Line, and Plane Postulates 17:24
   Intro 0:00 
   What are Postulates? 0:09 
    Definition of Postulates 0:10 
   Postulates 1:22 
    Postulate 1: Two Points 1:23 
    Postulate 2: Three Points 2:02 
    Postulate 3: Line 2:45 
   Postulates, cont.. 3:08 
    Postulate 4: Plane 3:09 
    Postulate 5: Two Points in a Plane 3:53 
   Postulates, cont.. 4:46 
    Postulate 6: Two Lines Intersect 4:47 
    Postulate 7: Two Plane Intersect 5:28 
   Using the Postulates 6:34 
    Examples: True or False 6:35 
   Using the Postulates 10:18 
    Examples: True or False 10:19 
   Extra Example 1: Always, Sometimes, or Never 12:22 
   Extra Example 2: Always, Sometimes, or Never 13:15 
   Extra Example 3: Always, Sometimes, or Never 14:16 
   Extra Example 4: Always, Sometimes, or Never 15:03 
  Deductive Reasoning 36:03
   Intro 0:00 
   Deductive Reasoning 0:06 
    Definition of Deductive Reasoning 0:07 
   Inductive vs. Deductive 2:51 
    Inductive Reasoning 2:52 
    Deductive reasoning 3:19 
   Law of Detachment 3:47 
    Law of Detachment 3:48 
    Examples of Law of Detachment 4:31 
   Law of Syllogism 7:32 
    Law of Syllogism 7:33 
    Example 1: Making a Conclusion 9:02 
    Example 2: Making a Conclusion 12:54 
   Using Laws of Logic 14:12 
    Example 1: Determine the Logic 14:42 
    Example 2: Determine the Logic 17:02 
   Using Laws of Logic, cont. 18:47 
    Example 3: Determine the Logic 19:03 
    Example 4: Determine the Logic 20:56 
   Extra Example 1: Determine the Conclusion and Law 22:12 
   Extra Example 2: Determine the Conclusion and Law 25:39 
   Extra Example 3: Determine the Logic and Law 29:50 
   Extra Example 4: Determine the Logic and Law 31:27 
  Proofs in Algebra: Properties of Equality 44:31
   Intro 0:00 
   Properties of Equality 0:10 
    Addition Property of Equality 0:28 
    Subtraction Property of Equality 1:10 
    Multiplication Property of Equality 1:41 
    Division Property of Equality 1:55 
    Addition Property of Equality Using Angles 2:46 
   Properties of Equality, cont. 4:10 
    Reflexive Property of Equality 4:11 
    Symmetric Property of Equality 5:24 
    Transitive Property of Equality 6:10 
   Properties of Equality, cont. 7:04 
    Substitution Property of Equality 7:05 
    Distributive Property of Equality 8:34 
   Two Column Proof 9:40 
    Example: Two Column Proof 9:46 
   Proof Example 1 16:13 
   Proof Example 2 23:49 
   Proof Example 3 30:33 
   Extra Example 1: Name the Property of Equality 38:07 
   Extra Example 2: Name the Property of Equality 40:16 
   Extra Example 3: Name the Property of Equality 41:35 
   Extra Example 4: Name the Property of Equality 43:02 
  Proving Segment Relationship 41:02
   Intro 0:00 
   Good Proofs 0:12 
    Five Essential Parts 0:13 
   Proof Reasons 1:38 
    Undefined 1:40 
    Definitions 2:06 
    Postulates 2:42 
    Previously Proven Theorems 3:24 
   Congruence of Segments 4:10 
    Theorem: Congruence of Segments 4:12 
   Proof Example 10:16 
    Proof: Congruence of Segments 10:17 
   Setting Up Proofs 19:13 
    Example: Two Segments with Equal Measures 19:15 
   Setting Up Proofs 21:48 
    Example: Vertical Angles are Congruent 21:50 
   Setting Up Proofs 23:59 
    Example: Segment of a Triangle 24:00 
   Extra Example 1: Congruence of Segments 27:03 
   Extra Example 2: Setting Up Proofs 28:50 
   Extra Example 3: Setting Up Proofs 30:55 
   Extra Example 4: Two-Column Proof 33:11 
  Proving Angle Relationships 33:37
   Intro 0:00 
   Supplement Theorem 0:05 
    Supplementary Angles 0:06 
   Congruence of Angles 2:37 
    Proof: Congruence of Angles 2:38 
   Angle Theorems 6:54 
    Angle Theorem 1: Supplementary Angles 6:55 
    Angle Theorem 2: Complementary Angles 10:25 
   Angle Theorems 11:32 
    Angle Theorem 3: Right Angles 11:35 
    Angle Theorem 4: Vertical Angles 12:09 
    Angle Theorem 5: Perpendicular Lines 12:57 
   Using Angle Theorems 13:45 
    Example 1: Always, Sometimes, or Never 13:50 
    Example 2: Always, Sometimes, or Never 14:28 
    Example 3: Always, Sometimes, or Never 16:21 
   Extra Example 1: Always, Sometimes, or Never 16:53 
   Extra Example 2: Find the Measure of Each Angle 18:55 
   Extra Example 3: Find the Measure of Each Angle 25:03 
   Extra Example 4: Two-Column Proof 27:08 
III. Perpendicular & Parallel Lines
  Parallel Lines and Transversals 37:35
   Intro 0:00 
   Lines 0:06 
    Parallel Lines 0:09 
    Skew Lines 2:02 
    Transversal 3:42 
   Angles Formed by a Transversal 4:28 
    Interior Angles 5:53 
    Exterior Angles 6:09 
    Consecutive Interior Angles 7:04 
    Alternate Exterior Angles 9:47 
    Alternate Interior Angles 11:22 
    Corresponding Angles 12:27 
   Angles Formed by a Transversal 15:29 
    Relationship Between Angles 15:30 
   Extra Example 1: Intersecting, Parallel, or Skew 19:26 
   Extra Example 2: Draw a Diagram 21:37 
   Extra Example 3: Name the Figures 24:12 
   Extra Example 4: Angles Formed by a Transversal 28:38 
  Angles and Parallel Lines 41:53
   Intro 0:00 
   Corresponding Angles Postulate 0:05 
    Corresponding Angles Postulate 0:06 
   Alternate Interior Angles Theorem 3:05 
    Alternate Interior Angles Theorem 3:07 
   Consecutive Interior Angles Theorem 5:16 
    Consecutive Interior Angles Theorem 5:17 
   Alternate Exterior Angles Theorem 6:42 
    Alternate Exterior Angles Theorem 6:43 
   Parallel Lines Cut by a Transversal 7:18 
    Example: Parallel Lines Cut by a Transversal 7:19 
   Perpendicular Transversal Theorem 14:54 
    Perpendicular Transversal Theorem 14:55 
   Extra Example 1: State the Postulate or Theorem 16:37 
   Extra Example 2: Find the Measure of the Numbered Angle 18:53 
   Extra Example 3: Find the Measure of Each Angle 25:13 
   Extra Example 4: Find the Values of x, y, and z 36:26 
  Slope of Lines 44:06
   Intro 0:00 
   Definition of Slope 0:06 
    Slope Equation 0:13 
   Slope of a Line 3:45 
    Example: Find the Slope of a Line 3:47 
   Slope of a Line 8:38 
    More Example: Find the Slope of a Line 8:40 
   Slope Postulates 12:32 
    Proving Slope Postulates 12:33 
   Parallel or Perpendicular Lines 17:23 
    Example: Parallel or Perpendicular Lines 17:24 
   Using Slope Formula 20:02 
    Example: Using Slope Formula 20:03 
   Extra Example 1: Slope of a Line 25:10 
   Extra Example 2: Slope of a Line 26:31 
   Extra Example 3: Graph the Line 34:11 
   Extra Example 4: Using the Slope Formula 38:50 
  Proving Lines Parallel 25:55
   Intro 0:00 
   Postulates 0:06 
    Postulate 1: Parallel Lines 0:21 
    Postulate 2: Parallel Lines 2:16 
   Parallel Postulate 3:28 
    Definition and Example of Parallel Postulate 3:29 
   Theorems 4:29 
    Theorem 1: Parallel Lines 4:40 
    Theorem 2: Parallel Lines 5:37 
   Theorems, cont. 6:10 
    Theorem 3: Parallel Lines 6:11 
   Extra Example 1: Determine Parallel Lines 6:56 
   Extra Example 2: Find the Value of x 11:42 
   Extra Example 3: Opposite Sides are Parallel 14:48 
   Extra Example 4: Proving Parallel Lines 20:42 
  Parallels and Distance 19:48
   Intro 0:00 
   Distance Between a Points and Line 0:07 
    Definition and Example 0:08 
   Distance Between Parallel Lines 1:51 
    Definition and Example 1:52 
   Extra Example 1: Drawing a Segment to Represent Distance 3:02 
   Extra Example 2: Drawing a Segment to Represent Distance 4:27 
   Extra Example 3: Graph, Plot, and Construct a Perpendicular Segment 5:13 
   Extra Example 4: Distance Between Two Parallel Lines 15:37 
IV. Congruent Triangles
  Classifying Triangles 28:43
   Intro 0:00 
   Triangles 0:09 
    Triangle: A Three-Sided Polygon 0:10 
    Sides 1:00 
    Vertices 1:22 
    Angles 1:56 
   Classifying Triangles by Angles 2:59 
    Acute Triangle 3:19 
    Obtuse Triangle 4:08 
    Right Triangle 4:44 
   Equiangular Triangle 5:38 
    Definition and Example of an Equiangular Triangle 5:39 
   Classifying Triangles by Sides 6:57 
    Scalene Triangle 7:17 
    Isosceles Triangle 7:57 
    Equilateral Triangle 8:12 
   Isosceles Triangle 8:58 
    Labeling Isosceles Triangle 9:00 
    Labeling Right Triangle 10:44 
   Isosceles Triangle 11:10 
    Example: Find x, AB, BC, and AC 11:11 
   Extra Example 1: Classify Each Triangle 13:45 
   Extra Example 2: Always, Sometimes, or Never 16:28 
   Extra Example 3: Find All the Sides of the Isosceles Triangle 20:29 
   Extra Example 4: Distance Formula and Triangle 22:29 
  Measuring Angles in Triangles 44:43
   Intro 0:00 
   Angle Sum Theorem 0:09 
    Angle Sum Theorem for Triangle 0:11 
   Using Angle Sum Theorem 4:06 
    Find the Measure of the Missing Angle 4:07 
   Third Angle Theorem 4:58 
    Example: Third Angle Theorem 4:59 
   Exterior Angle Theorem 7:58 
    Example: Exterior Angle Theorem 8:00 
   Flow Proof of Exterior Angle Theorem 15:14 
    Flow Proof of Exterior Angle Theorem 15:17 
   Triangle Corollaries 27:21 
    Triangle Corollary 1 27:50 
    Triangle Corollary 2 30:42 
   Extra Example 1: Find the Value of x 32:55 
   Extra Example 2: Find the Value of x 34:20 
   Extra Example 3: Find the Measure of the Angle 35:38 
   Extra Example 4: Find the Measure of Each Numbered Angle 39:00 
  Exploring Congruent Triangles 26:46
   Intro 0:00 
   Congruent Triangles 0:15 
    Example of Congruent Triangles 0:17 
   Corresponding Parts 3:39 
    Corresponding Angles and Sides of Triangles 3:40 
   Definition of Congruent Triangles 11:24 
    Definition of Congruent Triangles 11:25 
   Triangle Congruence 16:37 
    Congruence of Triangles 16:38 
   Extra Example 1: Congruence Statement 18:24 
   Extra Example 2: Congruence Statement 21:26 
   Extra Example 3: Draw and Label the Figure 23:09 
   Extra Example 4: Drawing Triangles 24:04 
  Proving Triangles Congruent 47:51
   Intro 0:00 
   SSS Postulate 0:18 
    Side-Side-Side Postulate 0:27 
   SAS Postulate 2:26 
    Side-Angle-Side Postulate 2:29 
   SAS Postulate 3:57 
    Proof Example 3:58 
   ASA Postulate 11:47 
    Angle-Side-Angle Postulate 11:53 
   AAS Theorem 14:13 
    Angle-Angle-Side Theorem 14:14 
   Methods Overview 16:16 
    Methods Overview 16:17 
    SSS 16:33 
    SAS 17:06 
    ASA 17:50 
    AAS 18:17 
    CPCTC 19:14 
   Extra Example 1:Proving Triangles are Congruent 21:29 
   Extra Example 2: Proof 25:40 
   Extra Example 3: Proof 30:41 
   Extra Example 4: Proof 38:41 
  Isosceles and Equilateral Triangles 27:53
   Intro 0:00 
   Isosceles Triangle Theorem 0:07 
    Isosceles Triangle Theorem 0:09 
   Isosceles Triangle Theorem 2:26 
    Example: Using the Isosceles Triangle Theorem 2:27 
   Isosceles Triangle Theorem Converse 3:29 
    Isosceles Triangle Theorem Converse 3:30 
   Equilateral Triangle Theorem Corollaries 4:30 
    Equilateral Triangle Theorem Corollary 1 4:59 
    Equilateral Triangle Theorem Corollary 2 5:55 
   Extra Example 1: Find the Value of x 7:08 
   Extra Example 2: Find the Value of x 10:04 
   Extra Example 3: Proof 14:04 
   Extra Example 4: Proof 22:41 
V. Triangle Inequalities
  Special Segments in Triangles 43:44
   Intro 0:00 
   Perpendicular Bisector 0:06 
    Perpendicular Bisector 0:07 
   Perpendicular Bisector 4:07 
    Perpendicular Bisector Theorems 4:08 
   Median 6:30 
    Definition of Median 6:31 
   Median 9:41 
    Example: Median 9:42 
   Altitude 12:22 
    Definition of Altitude 12:23 
   Angle Bisector 14:33 
    Definition of Angle Bisector 14:34 
   Angle Bisector 16:41 
    Angle Bisector Theorems 16:42 
   Special Segments Overview 18:57 
    Perpendicular Bisector 19:04 
    Median 19:32 
    Altitude 19:49 
    Angle Bisector 20:02 
    Examples: Special Segments 20:18 
   Extra Example 1: Draw and Label 22:36 
   Extra Example 2: Draw the Altitudes for Each Triangle 24:37 
   Extra Example 3: Perpendicular Bisector 27:57 
   Extra Example 4: Draw, Label, and Write Proof 34:33 
  Right Triangles 26:34
   Intro 0:00 
   LL Theorem 0:21 
    Leg-Leg Theorem 0:25 
   HA Theorem 2:23 
    Hypotenuse-Angle Theorem 2:24 
   LA Theorem 4:49 
    Leg-Angle Theorem 4:50 
   LA Theorem 6:18 
    Example: Find x and y 6:19 
   HL Postulate 8:22 
    Hypotenuse-Leg Postulate 8:23 
   Extra Example 1: LA Theorem & HL Postulate 10:57 
   Extra Example 2: Find x So That Each Pair of Triangles is Congruent 14:15 
   Extra Example 3: Two-column Proof 17:02 
   Extra Example 4: Two-column Proof 21:01 
  Indirect Proofs and Inequalities 33:30
   Intro 0:00 
   Writing an Indirect Proof 0:09 
    Step 1 0:49 
    Step 2 2:32 
    Step 3 3:00 
   Indirect Proof 4:30 
    Example: 2 + 6 = 8 5:00 
    Example: The Suspect is Guilty 5:40 
    Example: Measure of Angle A < Measure of Angle B 6:06 
   Definition of Inequality 7:47 
    Definition of Inequality & Example 7:48 
   Properties of Inequality 9:55 
    Comparison Property 9:58 
    Transitive Property 10:33 
    Addition and Subtraction Properties 12:01 
    Multiplication and Division Properties 13:07 
   Exterior Angle Inequality Theorem 14:12 
    Example: Exterior Angle Inequality Theorem 14:13 
   Extra Example 1: Draw a Diagram for the Statement 18:32 
   Extra Example 2: Name the Property for Each Statement 19:56 
   Extra Example 3: State the Assumption 21:22 
   Extra Example 4: Write an Indirect Proof 25:39 
  Inequalities for Sides and Angles of a Triangle 17:26
   Intro 0:00 
   Side to Angles 0:10 
    If One Side of a Triangle is Longer Than Another Side 0:11 
   Converse: Angles to Sides 1:57 
    If One Angle of a Triangle Has a Greater Measure Than Another Angle 1:58 
   Extra Example 1: Name the Angles in the Triangle From Least to Greatest 2:38 
   Extra Example 2: Find the Longest and Shortest Segment in the Triangle 3:47 
   Extra Example 3: Angles and Sides of a Triangle 4:51 
   Extra Example 4: Two-column Proof 9:08 
  Triangle Inequality 28:11
   Intro 0:00 
   Triangle Inequality Theorem 0:05 
    Triangle Inequality Theorem 0:06 
   Triangle Inequality Theorem 4:22 
    Example 1: Triangle Inequality Theorem 4:23 
    Example 2: Triangle Inequality Theorem 9:40 
   Extra Example 1: Determine if the Three Numbers can Represent the Sides of a Triangle 12:00 
   Extra Example 2: Finding the Third Side of a Triangle 13:34 
   Extra Example 3: Always True, Sometimes True, or Never True 18:18 
   Extra Example 4: Triangle and Vertices 22:36 
  Inequalities Involving Two Triangles 29:36
   Intro 0:00 
   SAS Inequality Theorem 0:06 
    SAS Inequality Theorem & Example 0:25 
   SSS Inequality Theorem 4:33 
    SSS Inequality Theorem & Example 4:34 
   Extra Example 1: Write an Inequality Comparing the Segments 6:08 
   Extra Example 2: Determine if the Statement is True 9:52 
   Extra Example 3: Write an Inequality for x 14:20 
   Extra Example 4: Two-column Proof 17:44 
VI. Quadrilaterals
  Parallelograms 29:11
   Intro 0:00 
   Quadrilaterals 0:06 
    Four-sided Polygons 0:08 
    Non Examples of Quadrilaterals 0:47 
   Parallelograms 1:35 
    Parallelograms 1:36 
   Properties of Parallelograms 4:28 
    Opposite Sides of a Parallelogram are Congruent 4:29 
    Opposite Angles of a Parallelogram are Congruent 5:49 
   Angles and Diagonals 6:24 
    Consecutive Angles in a Parallelogram are Supplementary 6:25 
    The Diagonals of a Parallelogram Bisect Each Other 8:42 
   Extra Example 1: Complete Each Statement About the Parallelogram 10:26 
   Extra Example 2: Find the Values of x, y, and z of the Parallelogram 13:21 
   Extra Example 3: Find the Distance of Each Side to Verify the Parallelogram 16:35 
   Extra Example 4: Slope of Parallelogram 23:15 
  Proving Parallelograms 42:43
   Intro 0:00 
   Parallelogram Theorems 0:09 
    Theorem 1 0:20 
    Theorem 2 1:50 
   Parallelogram Theorems, Cont. 3:10 
    Theorem 3 3:11 
    Theorem 4 4:15 
   Proving Parallelogram 6:21 
    Example: Determine if Quadrilateral ABCD is a Parallelogram 6:22 
   Summary 14:01 
    Both Pairs of Opposite Sides are Parallel 14:14 
    Both Pairs of Opposite Sides are Congruent 15:09 
    Both Pairs of Opposite Angles are Congruent 15:24 
    Diagonals Bisect Each Other 15:44 
    A Pair of Opposite Sides is Both Parallel and Congruent 16:13 
   Extra Example 1: Determine if Each Quadrilateral is a Parallelogram 16:54 
   Extra Example 2: Find the Value of x and y 20:23 
   Extra Example 3: Determine if the Quadrilateral ABCD is a Parallelogram 24:05 
   Extra Example 4: Two-column Proof 30:28 
  Rectangles 29:47
   Intro 0:00 
   Rectangles 0:03 
    Definition of Rectangles 0:04 
   Diagonals of Rectangles 2:52 
    Rectangles: Diagonals Property 1 2:53 
    Rectangles: Diagonals Property 2 3:30 
   Proving a Rectangle 4:40 
    Example: Determine Whether Parallelogram ABCD is a Rectangle 4:41 
   Rectangles Summary 9:22 
    Opposite Sides are Congruent and Parallel 9:40 
    Opposite Angles are Congruent 9:51 
    Consecutive Angles are Supplementary 9:58 
    Diagonals are Congruent and Bisect Each Other 10:05 
    All Four Angles are Right Angles 10:40 
   Extra Example 1: Find the Value of x 11:03 
   Extra Example 2: Name All Congruent Sides and Angles 13:52 
   Extra Example 3: Always, Sometimes, or Never True 19:39 
   Extra Example 4: Determine if ABCD is a Rectangle 26:45 
  Squares and Rhombi 39:14
   Intro 0:00 
   Rhombus 0:09 
    Definition of a Rhombus 0:10 
   Diagonals of a Rhombus 2:03 
    Rhombus: Diagonals Property 1 2:21 
    Rhombus: Diagonals Property 2 3:49 
    Rhombus: Diagonals Property 3 4:36 
   Rhombus 6:17 
    Example: Use the Rhombus to Find the Missing Value 6:18 
   Square 8:17 
    Definition of a Square 8:20 
   Summary Chart 11:06 
    Parallelogram 11:07 
    Rectangle 12:56 
    Rhombus 13:54 
    Square 14:44 
   Extra Example 1: Diagonal Property 15:44 
   Extra Example 2: Use Rhombus ABCD to Find the Missing Value 19:39 
   Extra Example 3: Always, Sometimes, or Never True 23:06 
   Extra Example 4: Determine the Quadrilateral 28:02 
  Trapezoids and Kites 30:48
   Intro 0:00 
   Trapezoid 0:10 
    Definition of Trapezoid 0:12 
   Isosceles Trapezoid 2:57 
    Base Angles of an Isosceles Trapezoid 2:58 
    Diagonals of an Isosceles Trapezoid 4:05 
   Median of a Trapezoid 4:26 
    Median of a Trapezoid 4:27 
   Median of a Trapezoid 6:41 
    Median Formula 7:00 
   Kite 8:28 
    Definition of a Kite 8:29 
   Quadrilaterals Summary 11:19 
    A Quadrilateral with Two Pairs of Adjacent Congruent Sides 11:20 
   Extra Example 1: Isosceles Trapezoid 14:50 
   Extra Example 2: Median of Trapezoid 18:28 
   Extra Example 3: Always, Sometimes, or Never 24:13 
   Extra Example 4: Determine if the Figure is a Trapezoid 26:49 
VII. Proportions and Similarity
  Using Proportions and Ratios 20:10
   Intro 0:00 
   Ratio 0:05 
    Definition and Examples of Writing Ratio 0:06 
   Proportion 2:05 
    Definition of Proportion 2:06 
    Examples of Proportion 2:29 
   Using Ratio 5:53 
    Example: Ratio 5:54 
   Extra Example 1: Find Three Ratios Equivalent to 2/5 9:28 
   Extra Example 2: Proportion and Cross Products 10:32 
   Extra Example 3: Express Each Ratio as a Fraction 13:18 
   Extra Example 4: Fin the Measure of a 3:4:5 Triangle 17:26 
  Similar Polygons 27:53
   Intro 0:00 
   Similar Polygons 0:05 
    Definition of Similar Polygons 0:06 
    Example of Similar Polygons 2:32 
   Scale Factor 4:26 
    Scale Factor: Definition and Example 4:27 
   Extra Example 1: Determine if Each Pair of Figures is Similar 7:03 
   Extra Example 2: Find the Values of x and y 11:33 
   Extra Example 3: Similar Triangles 19:57 
   Extra Example 4: Draw Two Similar Figures 23:36 
  Similar Triangles 34:10
   Intro 0:00 
   AA Similarity 0:10 
    Definition of AA Similarity 0:20 
    Example of AA Similarity 2:32 
   SSS Similarity 4:46 
    Definition of SSS Similarity 4:47 
    Example of SSS Similarity 6:00 
   SAS Similarity 8:04 
    Definition of SAS Similarity 8:05 
    Example of SAS Similarity 9:12 
   Extra Example 1: Determine Whether Each Pair of Triangles is Similar 10:59 
   Extra Example 2: Determine Which Triangles are Similar 16:08 
   Extra Example 3: Determine if the Statement is True or False 23:11 
   Extra Example 4: Write Two-Column Proof 26:25 
  Parallel Lines and Proportional Parts 24:07
   Intro 0:00 
   Triangle Proportionality 0:07 
    Definition of Triangle Proportionality 0:08 
    Example of Triangle Proportionality 0:51 
   Triangle Proportionality Converse 2:19 
    Triangle Proportionality Converse 2:20 
   Triangle Mid-segment 3:42 
    Triangle Mid-segment: Definition and Example 3:43 
   Parallel Lines and Transversal 6:51 
    Parallel Lines and Transversal 6:52 
   Extra Example 1: Complete Each Statement 8:59 
   Extra Example 2: Determine if the Statement is True or False 12:28 
   Extra Example 3: Find the Value of x and y 15:35 
   Extra Example 4: Find Midpoints of a Triangle 20:43 
  Parts of Similar Triangles 27:06
   Intro 0:00 
   Proportional Perimeters 0:09 
    Proportional Perimeters: Definition and Example 0:10 
   Similar Altitudes 2:23 
    Similar Altitudes: Definition and Example 2:24 
   Similar Angle Bisectors 4:50 
    Similar Angle Bisectors: Definition and Example 4:51 
   Similar Medians 6:05 
    Similar Medians: Definition and Example 6:06 
   Angle Bisector Theorem 7:33 
    Angle Bisector Theorem 7:34 
   Extra Example 1: Parts of Similar Triangles 10:52 
   Extra Example 2: Parts of Similar Triangles 14:57 
   Extra Example 3: Parts of Similar Triangles 19:27 
   Extra Example 4: Find the Perimeter of Triangle ABC 23:14 
VIII. Applying Right Triangles & Trigonometry
  Pythagorean Theorem 21:14
   Intro 0:00 
   Pythagorean Theorem 0:05 
    Pythagorean Theorem & Example 0:06 
   Pythagorean Converse 1:20 
    Pythagorean Converse & Example 1:21 
   Pythagorean Triple 2:42 
    Pythagorean Triple 2:43 
   Extra Example 1: Find the Missing Side 4:59 
   Extra Example 2: Determine Right Triangle 7:40 
   Extra Example 3: Determine Pythagorean Triple 11:30 
   Extra Example 4: Vertices and Right Triangle 14:29 
  Geometric Mean 40:59
   Intro 0:00 
   Geometric Mean 0:04 
    Geometric Mean & Example 0:05 
   Similar Triangles 4:32 
    Similar Triangles 4:33 
   Geometric Mean-Altitude 11:10 
    Geometric Mean-Altitude & Example 11:11 
   Geometric Mean-Leg 14:47 
    Geometric Mean-Leg & Example 14:18 
   Extra Example 1: Geometric Mean Between Each Pair of Numbers 20:10 
   Extra Example 2: Similar Triangles 23:46 
   Extra Example 3: Geometric Mean of Triangles 28:30 
   Extra Example 4: Geometric Mean of Triangles 36:58 
  Special Right Triangles 37:57
   Intro 0:00 
   45-45-90 Triangles 0:06 
    Definition of 45-45-90 Triangles 0:25 
   45-45-90 Triangles 5:51 
    Example: Find n 5:52 
   30-60-90 Triangles 8:59 
    Definition of 30-60-90 Triangles 9:00 
   30-60-90 Triangles 12:25 
    Example: Find n 12:26 
   Extra Example 1: Special Right Triangles 15:08 
   Extra Example 2: Special Right Triangles 18:22 
   Extra Example 3: Word Problems & Special Triangles 27:40 
   Extra Example 4: Hexagon & Special Triangles 33:51 
  Ratios in Right Triangles 40:37
   Intro 0:00 
   Trigonometric Ratios 0:08 
    Definition of Trigonometry 0:13 
    Sine (sin), Cosine (cos), & Tangent (tan) 0:50 
   Trigonometric Ratios 3:04 
    Trig Functions 3:05 
    Inverse Trig Functions 5:02 
   SOHCAHTOA 8:16 
    sin x 9:07 
    cos x 10:00 
    tan x 10:32 
    Example: SOHCAHTOA & Triangle 12:10 
   Extra Example 1: Find the Value of Each Ratio or Angle Measure 14:36 
   Extra Example 2: Find Sin, Cos, and Tan 18:51 
   Extra Example 3: Find the Value of x Using SOHCAHTOA 22:55 
   Extra Example 4: Trigonometric Ratios in Right Triangles 32:13 
  Angles of Elevation and Depression 21:04
   Intro 0:00 
   Angle of Elevation 0:10 
    Definition of Angle of Elevation & Example 0:11 
   Angle of Depression 1:19 
    Definition of Angle of Depression & Example 1:20 
   Extra Example 1: Name the Angle of Elevation and Depression 2:22 
   Extra Example 2: Word Problem & Angle of Depression 4:41 
   Extra Example 3: Word Problem & Angle of Elevation 14:02 
   Extra Example 4: Find the Missing Measure 18:10 
  Law of Sines 35:25
   Intro 0:00 
   Law of Sines 0:20 
    Law of Sines 0:21 
   Law of Sines 3:34 
    Example: Find b 3:35 
   Solving the Triangle 9:19 
    Example: Using the Law of Sines to Solve Triangle 9:20 
   Extra Example 1: Law of Sines and Triangle 17:43 
   Extra Example 2: Law of Sines and Triangle 20:06 
   Extra Example 3: Law of Sines and Triangle 23:54 
   Extra Example 4: Law of Sines and Triangle 28:59 
  Law of Cosines 52:43
   Intro 0:00 
   Law of Cosines 0:35 
    Law of Cosines 0:36 
   Law of Cosines 6:22 
    Use the Law of Cosines When Both are True 6:23 
   Law of Cosines 8:35 
    Example: Law of Cosines 8:36 
   Extra Example 1: Law of Sines or Law of Cosines? 13:35 
   Extra Example 2: Use the Law of Cosines to Find the Missing Measure 17:02 
   Extra Example 3: Solve the Triangle 30:49 
   Extra Example 4: Find the Measure of Each Diagonal of the Parallelogram 41:39 
IX. Circles
  Segments in a Circle 22:43
   Intro 0:00 
   Segments in a Circle 0:10 
    Circle 0:11 
    Chord 0:59 
    Diameter 1:32 
    Radius 2:07 
    Secant 2:17 
    Tangent 3:10 
   Circumference 3:56 
    Introduction to Circumference 3:57 
    Example: Find the Circumference of the Circle 5:09 
   Circumference 6:40 
    Example: Find the Circumference of the Circle 6:41 
   Extra Example 1: Use the Circle to Answer the Following 9:10 
   Extra Example 2: Find the Missing Measure 12:53 
   Extra Example 3: Given the Circumference, Find the Perimeter of the Triangle 15:51 
   Extra Example 4: Find the Circumference of Each Circle 19:24 
  Angles and Arc 35:24
   Intro 0:00 
   Central Angle 0:06 
    Definition of Central Angle 0:07 
   Sum of Central Angles 1:17 
    Sum of Central Angles 1:18 
   Arcs 2:27 
    Minor Arc 2:30 
    Major Arc 3:47 
   Arc Measure 5:24 
    Measure of Minor Arc 5:24 
    Measure of Major Arc 6:53 
    Measure of a Semicircle 7:11 
   Arc Addition Postulate 8:25 
    Arc Addition Postulate 8:26 
   Arc Length 9:43 
    Arc Length and Example 9:44 
   Concentric Circles 16:05 
    Concentric Circles 16:06 
   Congruent Circles and Arcs 17:50 
    Congruent Circles 17:51 
    Congruent Arcs 18:47 
   Extra Example 1: Minor Arc, Major Arc, and Semicircle 20:14 
   Extra Example 2: Measure and Length of Arc 22:52 
   Extra Example 3: Congruent Arcs 25:48 
   Extra Example 4: Angles and Arcs 30:33 
  Arcs and Chords 21:51
   Intro 0:00 
   Arcs and Chords 0:07 
    Arc of the Chord 0:08 
    Theorem 1: Congruent Minor Arcs 1:01 
   Inscribed Polygon 2:10 
    Inscribed Polygon 2:11 
   Arcs and Chords 3:18 
    Theorem 2: When a Diameter is Perpendicular to a Chord 3:19 
   Arcs and Chords 5:05 
    Theorem 3: Congruent Chords 5:06 
   Extra Example 1: Congruent Arcs 10:35 
   Extra Example 2: Length of Arc 13:50 
   Extra Example 3: Arcs and Chords 17:09 
   Extra Example 4: Arcs and Chords 19:45 
  Inscribed Angles 27:53
   Intro 0:00 
   Inscribed Angles 0:07 
    Definition of Inscribed Angles 0:08 
   Inscribed Angles 0:58 
    Inscribed Angle Theorem 1 0:59 
   Inscribed Angles 3:29 
    Inscribed Angle Theorem 2 3:30 
   Inscribed Angles 4:38 
    Inscribed Angle Theorem 3 4:39 
   Inscribed Quadrilateral 5:50 
    Inscribed Quadrilateral 5:51 
   Extra Example 1: Central Angle, Inscribed Angle, and Intercepted Arc 7:02 
   Extra Example 2: Inscribed Angles 9:24 
   Extra Example 3: Inscribed Angles 14:00 
   Extra Example 4: Complete the Proof 17:58 
  Tangents 26:16
   Intro 0:00 
   Tangent Theorems 0:04 
    Tangent Theorem 1 0:05 
    Tangent Theorem 1 Converse 0:55 
   Common Tangents 1:34 
    Common External Tangent 2:12 
    Common Internal Tangent 2:30 
   Tangent Segments 3:08 
    Tangent Segments 3:09 
   Circumscribed Polygons 4:11 
    Circumscribed Polygons 4:12 
   Extra Example 1: Tangents & Circumscribed Polygons 5:50 
   Extra Example 2: Tangents & Circumscribed Polygons 8:35 
   Extra Example 3: Tangents & Circumscribed Polygons 11:50 
   Extra Example 4: Tangents & Circumscribed Polygons 15:43 
  Secants, Tangents, & Angle Measures 27:50
   Intro 0:00 
   Secant 0:08 
    Secant 0:09 
   Secant and Tangent 0:49 
    Secant and Tangent 0:50 
   Interior Angles 2:56 
    Secants & Interior Angles 2:57 
   Exterior Angles 7:21 
    Secants & Exterior Angles 7:22 
   Extra Example 1: Secants, Tangents, & Angle Measures 10:53 
   Extra Example 2: Secants, Tangents, & Angle Measures 13:31 
   Extra Example 3: Secants, Tangents, & Angle Measures 19:54 
   Extra Example 4: Secants, Tangents, & Angle Measures 22:29 
  Special Segments in a Circle 23:08
   Intro 0:00 
   Chord Segments 0:05 
    Chord Segments 0:06 
   Secant Segments 1:36 
    Secant Segments 1:37 
   Tangent and Secant Segments 4:10 
    Tangent and Secant Segments 4:11 
   Extra Example 1: Special Segments in a Circle 5:53 
   Extra Example 2: Special Segments in a Circle 7:58 
   Extra Example 3: Special Segments in a Circle 11:24 
   Extra Example 4: Special Segments in a Circle 18:09 
  Equations of Circles 27:01
   Intro 0:00 
   Equation of a Circle 0:06 
    Standard Equation of a Circle 0:07 
    Example 1: Equation of a Circle 0:57 
    Example 2: Equation of a Circle 1:36 
   Extra Example 1: Determine the Coordinates of the Center and the Radius 4:56 
   Extra Example 2: Write an Equation Based on the Given Information 7:53 
   Extra Example 3: Graph Each Circle 16:48 
   Extra Example 4: Write the Equation of Each Circle 19:17 
X. Polygons & Area
  Polygons 27:24
   Intro 0:00 
   Polygons 0:10 
    Polygon vs. Not Polygon 0:18 
   Convex and Concave 1:46 
    Convex vs. Concave Polygon 1:52 
   Regular Polygon 4:04 
    Regular Polygon 4:05 
   Interior Angle Sum Theorem 4:53 
    Triangle 5:03 
    Quadrilateral 6:05 
    Pentagon 6:38 
    Hexagon 7:59 
    20-Gon 9:36 
   Exterior Angle Sum Theorem 12:04 
    Exterior Angle Sum Theorem 12:05 
   Extra Example 1: Drawing Polygons 13:51 
   Extra Example 2: Convex Polygon 15:16 
   Extra Example 3: Exterior Angle Sum Theorem 18:21 
   Extra Example 4: Interior Angle Sum Theorem 22:20 
  Area of Parallelograms 17:46
   Intro 0:00 
   Parallelograms 0:06 
    Definition and Area Formula 0:07 
   Area of Figure 2:00 
    Area of Figure 2:01 
   Extra Example 1:Find the Area of the Shaded Area 3:14 
   Extra Example 2: Find the Height and Area of the Parallelogram 6:00 
   Extra Example 3: Find the Area of the Parallelogram Given Coordinates and Vertices 10:11 
   Extra Example 4: Find the Area of the Figure 14:31 
  Area of Triangles Rhombi, & Trapezoids 20:31
   Intro 0:00 
   Area of a Triangle 0:06 
    Area of a Triangle: Formula and Example 0:07 
   Area of a Trapezoid 2:31 
    Area of a Trapezoid: Formula 2:32 
    Area of a Trapezoid: Example 6:55 
   Area of a Rhombus 8:05 
    Area of a Rhombus: Formula and Example 8:06 
   Extra Example 1: Find the Area of the Polygon 9:51 
   Extra Example 2: Find the Area of the Figure 11:19 
   Extra Example 3: Find the Area of the Figure 14:16 
   Extra Example 4: Find the Height of the Trapezoid 18:10 
  Area of Regular Polygons & Circles 36:43
   Intro 0:00 
   Regular Polygon 0:08 
    SOHCAHTOA 0:54 
    30-60-90 Triangle 1:52 
    45-45-90 Triangle 2:40 
   Area of a Regular Polygon 3:39 
    Area of a Regular Polygon 3:40 
   Are of a Circle 7:55 
    Are of a Circle 7:56 
   Extra Example 1: Find the Area of the Regular Polygon 8:22 
   Extra Example 2: Find the Area of the Regular Polygon 16:48 
   Extra Example 3: Find the Area of the Shaded Region 24:11 
   Extra Example 4: Find the Area of the Shaded Region 32:24 
  Perimeter & Area of Similar Figures 18:17
   Intro 0:00 
   Perimeter of Similar Figures 0:08 
    Example: Scale Factor & Perimeter of Similar Figures 0:09 
   Area of Similar Figures 2:44 
    Example:Scale Factor & Area of Similar Figures 2:55 
   Extra Example 1: Complete the Table 6:09 
   Extra Example 2: Find the Ratios of the Perimeter and Area of the Similar Figures 8:56 
   Extra Example 3: Find the Unknown Area 12:04 
   Extra Example 4: Use the Given Area to Find AB 14:26 
  Geometric Probability 38:40
   Intro 0:00 
   Length Probability Postulate 0:05 
    Length Probability Postulate 0:06 
   Are Probability Postulate 2:34 
    Are Probability Postulate 2:35 
   Are of a Sector of a Circle 4:11 
    Are of a Sector of a Circle Formula 4:12 
    Are of a Sector of a Circle Example 7:51 
   Extra Example 1: Length Probability 11:07 
   Extra Example 2: Area Probability 12:14 
   Extra Example 3: Area Probability 17:17 
   Extra Example 4: Area of a Sector of a Circle 26:23 
XI. Solids
  Three-Dimensional Figures 23:39
   Intro 0:00 
   Polyhedrons 0:05 
    Polyhedrons: Definition and Examples 0:06 
    Faces 1:08 
    Edges 1:55 
    Vertices 2:23 
   Solids 2:51 
    Pyramid 2:54 
    Cylinder 3:45 
    Cone 4:09 
    Sphere 4:23 
   Prisms 5:00 
     Rectangular, Regular, and Cube Prisms 5:02 
   Platonic Solids 9:48 
    Five Types of Regular Polyhedra 9:49 
   Slices and Cross Sections 12:07 
    Slices 12:08 
    Cross Sections 12:47 
   Extra Example 1: Name the Edges, Faces, and Vertices of the Polyhedron 14:23 
   Extra Example 2: Determine if the Figure is a Polyhedron and Explain Why 17:37 
   Extra Example 3: Describe the Slice Resulting from the Cut 19:12 
   Extra Example 4: Describe the Shape of the Intersection 21:25 
  Surface Area of Prisms and Cylinders 38:50
   Intro 0:00 
   Prisms 0:06 
    Bases 0:07 
    Lateral Faces 0:52 
    Lateral Edges 1:19 
    Altitude 1:58 
   Prisms 2:24 
    Right Prism 2:25 
    Oblique Prism 2:56 
   Classifying Prisms 3:27 
    Right Rectangular Prism 3:28 
     4:55 
    Oblique Pentagonal Prism 6:26 
    Right Hexagonal Prism 7:14 
   Lateral Area of a Prism 7:42 
    Lateral Area of a Prism 7:43 
   Surface Area of a Prism 13:44 
    Surface Area of a Prism 13:45 
   Cylinder 16:18 
    Cylinder: Right and Oblique 16:19 
   Lateral Area of a Cylinder 18:02 
    Lateral Area of a Cylinder 18:03 
   Surface Area of a Cylinder 20:54 
    Surface Area of a Cylinder 20:55 
   Extra Example 1: Find the Lateral Area and Surface Are of the Prism 21:51 
   Extra Example 2: Find the Lateral Area of the Prism 28:15 
   Extra Example 3: Find the Surface Area of the Prism 31:57 
   Extra Example 4: Find the Lateral Area and Surface Area of the Cylinder 34:17 
  Surface Area of Pyramids and Cones 26:10
   Intro 0:00 
   Pyramids 0:07 
    Pyramids 0:08 
   Regular Pyramids 1:52 
    Regular Pyramids 1:53 
   Lateral Area of a Pyramid 4:33 
    Lateral Area of a Pyramid 4:34 
   Surface Area of a Pyramid 9:19 
    Surface Area of a Pyramid 9:20 
   Cone 10:09 
    Right and Oblique Cone 10:10 
   Lateral Area and Surface Area of a Right Cone 11:20 
    Lateral Area and Surface Are of a Right Cone 11:21 
   Extra Example 1: Pyramid and Prism 13:11 
   Extra Example 2: Find the Lateral Area of the Regular Pyramid 15:00 
   Extra Example 3: Find the Surface Area of the Pyramid 18:29 
   Extra Example 4: Find the Lateral Area and Surface Area of the Cone 22:08 
  Volume of Prisms and Cylinders 21:59
   Intro 0:00 
   Volume of Prism 0:08 
    Volume of Prism 0:10 
   Volume of Cylinder 3:38 
    Volume of Cylinder 3:39 
   Extra Example 1: Find the Volume of the Prism 5:10 
   Extra Example 2: Find the Volume of the Cylinder 8:03 
   Extra Example 3: Find the Volume of the Prism 9:35 
   Extra Example 4: Find the Volume of the Solid 19:06 
  Volume of Pyramids and Cones 22:02
   Intro 0:00 
   Volume of a Cone 0:08 
    Volume of a Cone: Example 0:10 
   Volume of a Pyramid 3:02 
    Volume of a Pyramid: Example 3:03 
   Extra Example 1: Find the Volume of the Pyramid 4:56 
   Extra Example 2: Find the Volume of the Solid 6:01 
   Extra Example 3: Find the Volume of the Pyramid 10:28 
   Extra Example 4: Find the Volume of the Octahedron 16:23 
  Surface Area and Volume of Spheres 14:46
   Intro 0:00 
   Special Segments 0:06 
    Radius 0:07 
    Chord 0:31 
    Diameter 0:55 
    Tangent 1:20 
   Sphere 1:43 
    Plane & Sphere 1:44 
    Hemisphere 2:56 
   Surface Area of a Sphere 3:25 
    Surface Area of a Sphere 3:26 
   Volume of a Sphere 4:08 
    Volume of a Sphere 4:09 
   Extra Example 1: Determine Whether Each Statement is True or False 4:24 
   Extra Example 2: Find the Surface Area of the Sphere 6:17 
   Extra Example 3: Find the Volume of the Sphere with a Diameter of 20 Meters 7:25 
   Extra Example 4: Find the Surface Area and Volume of the Solid 9:17 
  Congruent and Similar Solids 16:06
   Intro 0:00 
   Scale Factor 0:06 
    Scale Factor: Definition and Example 0:08 
   Congruent Solids 1:09 
    Congruent Solids 1:10 
   Similar Solids 2:17 
    Similar Solids 2:18 
   Extra Example 1: Determine if Each Pair of Solids is Similar, Congruent, or Neither 3:35 
   Extra Example 2: Determine if Each Statement is True or False 7:47 
   Extra Example 3: Find the Scale Factor and the Ratio of the Surface Areas and Volume 10:14 
   Extra Example 4: Find the Volume of the Larger Prism 12:14 
XII. Transformational Geometry
  Mapping 14:12
   Intro 0:00 
   Transformation 0:04 
    Rotation 0:32 
    Translation 1:03 
    Reflection 1:17 
    Dilation 1:24 
   Transformations 1:45 
    Examples 1:46 
   Congruence Transformation 2:51 
    Congruence Transformation 2:52 
   Extra Example 1: Describe the Transformation that Occurred in the Mappings 3:37 
   Extra Example 2: Determine if the Transformation is an Isometry 5:16 
   Extra Example 3: Isometry 8:16 
  Reflections 23:17
   Intro 0:00 
   Reflection 0:05 
    Definition of Reflection 0:06 
    Line of Reflection 0:35 
    Point of Reflection 1:22 
   Symmetry 1:59 
    Line of Symmetry 2:00 
    Point of Symmetry 2:48 
   Extra Example 1: Draw the Image over the Line of Reflection and the Point of Reflection 3:45 
   Extra Example 2: Determine Lines and Point of Symmetry 6:59 
   Extra Example 3: Graph the Reflection of the Polygon 11:15 
   Extra Example 4: Graph the Coordinates 16:07 
  Translations 18:43
   Intro 0:00 
   Translation 0:05 
    Translation: Preimage & Image 0:06 
    Example 0:56 
   Composite of Reflections 6:28 
    Composite of Reflections 6:29 
   Extra Example 1: Translation 7:48 
   Extra Example 2: Image, Preimage, and Translation 12:38 
   Extra Example 3: Find the Translation Image Using a Composite of Reflections 15:08 
   Extra Example 4: Find the Value of Each Variable in the Translation 17:18 
  Rotations 21:26
   Intro 0:00 
   Rotations 0:04 
    Rotations 0:05 
   Performing Rotations 2:13 
    Composite of Two Successive Reflections over Two Intersecting Lines 2:14 
    Angle of Rotation: Angle Formed by Intersecting Lines 4:29 
   Angle of Rotation 5:30 
    Rotation Postulate 5:31 
   Extra Example 1: Find the Rotated Image 7:32 
   Extra Example 2: Rotations and Coordinate Plane 10:33 
   Extra Example 3: Find the Value of Each Variable in the Rotation 14:29 
   Extra Example 4: Draw the Polygon Rotated 90 Degree Clockwise about P 16:13 
  Dilation 37:06
   Intro 0:00 
   Dilations 0:06 
    Dilations 0:07 
   Scale Factor 1:36 
    Scale Factor 1:37 
    Example 1 2:06 
    Example 2 6:22 
   Scale Factor 8:20 
    Positive Scale Factor 8:21 
    Negative Scale Factor 9:25 
    Enlargement 12:43 
    Reduction 13:52 
   Extra Example 1: Find the Scale Factor 16:39 
   Extra Example 2: Find the Measure of the Dilation Image 19:32 
   Extra Example 3: Find the Coordinates of the Image with Scale Factor and the Origin as the Center of Dilation 26:18 
   Extra Example 4: Graphing Polygon, Dilation, and Scale Factor 32:08