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Proving Lines Parallel

  • Postulates:
    • If two lines in a plane are cut by a transversal so that corresponding angles are congruent, then the lines are parallel
    • If two lines in a plane are cut by a transversal so that a pair of alternate exterior angles are congruent, then the two lines are parallel
    • Parallel Postulate: IF there is a line and a point not on the line, then there exists exactly one line through the point that is parallel to the given line
  • Theorems:
    • If two lines in a plane are cut by a transversal so that a pair of consecutive interior angles is supplementary, then the lines are parallel
    • If two lines in a plane are cut by a transversal so that a pair of alternate interior angles is congruent, then the lines are parallel
    • In a plane, if two lines are perpendicular to the same line, then they are parallel

Proving Lines Parallel

Use always, sometimes, and never to fill the blank of the statement to make it true.
If lines m and n are cut by a transversal line q and the corresponding angles are congruent, then line m is ____ parrallel to line n.
  
Always.
Determine the following statement is true or false.
Point A is not on line m, then there are more than one lines that pass through point A and are prallel to line m.
  
False.
∠1 ≅ ∠2, determine the relationship of line m and line n.
  
Line m is parallel to line n.
∠2 ≅ ∠4y, find a pair of parallel lines, and state which postulate or theorem is used.
  • line r and line p are parallel.
The theorem used: If two lines in a plane are cut by a transversal so that a pair of alternate interior angles is congruent, then the lines are parallel.
∠3 ≅ ∠9,find a pair of parallel lines, and state which postulate or theorem is used.
  • line n is parallel to line l.
the postulate used: If two lines in a plane are cut by a transversal so that corresponding angles are congruent, then the lines are parallel.
m∠8 + m∠9 = 180o, find a pair of parallel lines, and state which postulate or theorem is used.
  • line p is parallel to line q.
the theorem used: If two lines in a plane are cut by a transversal so that a pair of consecutive interior angles is supplementary, then the lines are parallel.
∠7 ≅ ∠12,find a pair of parallel lines, and state which postulate or theorem is used.
y
  • line p is parallel to line r.
The postulate used: If two lines in a plane are cut by a transversal so that a pair of alternate exterior angles is supplementary, then the lines are parallel.
p⊥m, q⊥m, find a pair of parallel lines, and state which postulate or theorem is used.
  • line p is parallel to line q.
The theorem used: In a plane, if two lines are perpendicular to the same line, then they are parallel.
m||n, m∠1 = 120o, find m∠3.
  • ∠2 ≅ ∠1
  • m∠2 = m∠1 = 120o
  • m∠2 + m∠3 = 180o
m∠3 = 180o − m∠2 = 180o − 120o = 60o.
m||n, m∠1 = x + 20, m∠2 = 2x + 4, find x.
  • ∠2 ≅ ∠1
  • m∠2 = m∠1
  • 2x + 4 = x + 20
x = 16

*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 Lines Parallel

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 Lines Parallel

Welcome back to Educator.com.0000

This next lesson is on proving lines parallel.0002

We are actually going to take the theorems that we learned from the past few lessons, and we are going to use them to prove that two lines are parallel.0007

We learned, from the Corresponding Angles Postulate, that if the lines are parallel, then the corresponding angles are congruent.0022

If the lines are the parallel lines that are cut by a transversal, then the corresponding angles are congruent.0035

Now, this one is saying, "If the two lines in a plane are cut by a transversal, and corresponding angles are congruent, then the lines are parallel."0045

So, it is using the converse of that postulate that we learned a couple of lessons ago.0053

And we are going to take that and use it to prove that lines are parallel.0058

Before, when we used that postulate, it was given that the lines were parallel; and then you would have to show0065

that the conclusion, "then the corresponding angles are congruent," would be true.0071

But for this one, they are giving you that the corresponding angles are congruent.0077

And then, the conclusion, "the lines are parallel," is what you are going to be proving.0083

This first postulate: if you look at angles 1 and 2, those are corresponding angles.0090

So, if I tell you that angle 1 and angle 2 are congruent, and they are corresponding angles, then the lines are parallel.0096

As long as these two angles are congruent, then these lines are parallel; so I can conclude that these lines are parallel.0104

Now, again, for them to be corresponding angles, the lines don't have to be parallel.0115

Even if the lines are not parallel, they are still considered corresponding angles.0124

But now, we know that, as long as the corresponding angles are congruent, then the lines will be parallel.0129

The next postulate: If two lines in a plane are cut by a transversal so that a pair of alternate exterior angles are congruent, then the two lines are parallel.0136

So again, this is the alternate exterior angles theorem's converse.0149

The alternate exterior angles theorem said that, if two lines are parallel, then alternate exterior angles are congruent.0156

This is the converse; so they are giving you that alternate exterior angles are congruent; then, the lines are parallel.0168

Depending on what you are trying to prove, you are going to be using the different postulates--either the original theorem or postulate, or the converse, these.0176

If you are trying to prove that the lines are parallel, then you are going to be using these.0188

If you are trying to prove that the angles are congruent, then you are going to be using the original.0191

So, alternate exterior angles are congruent; therefore, we can conclude that the lines are parallel.0197

The Parallel Postulate: this is called the Parallel Postulate: If there is a line and a point not on the line,0209

then there exists exactly one line through that point that is parallel to the given line.0220

What that is saying is that I could only draw one line, a single line, through this point, so that it is going to be parallel to this line.0227

I cannot draw two different lines and have them both be parallel to this line--only one.0239

So, it would look something like that...well, that is not really through the point, but...0246

this is the only line that I can draw to make it parallel to this line.0256

Only one line exists; and that is the Parallel Postulate.0262

Now, we are going to go over a few more theorems now.0270

Before it was postulates, but now these are some theorems that we can use to prove lines parallel.0273

If two lines in a plane are cut by a transversal (this is my transversal) so that a pair of consecutive interior angles is supplementary, then the lines are parallel.0281

Remember consecutive interior angles? They are not congruent; they are supplementary.0292

If you look at these angles, angle 1 is an obtuse angle; angle 2 is an acute angle.0297

They don't even look congruent; they don't look the same.0303

Make sure that consecutive interior angles are supplementary.0307

Again, the original theorem had said, "Well, if the lines are parallel" (that is given), "then we can conclude that consecutive interior angles are supplementary."0312

This one is the converse, saying that the given is that the consecutive interior angles are supplementary.0323

Then, the conclusion is...we can conclude that these lines are parallel.0330

If two lines in a plane are cut by the transversal, so that a pair of alternate interior angles is congruent, then the lines are parallel.0338

So again, if these alternate interior angles are congruent, then the lines are parallel.0350

And make sure that it is not the transversal; it is the two lines that the transversal is cutting through that are parallel.0363

And the next theorem, the last theorem that we are going to go over today for this lesson:0371

in a plane, if two lines are perpendicular to the same line, then they are parallel.0375

See how this line is perpendicular to this line? Well, this is my transversal.0383

So, if this line is perpendicular to this line, and this line is also perpendicular to that same line, the same transversal, then these lines will be parallel.0387

If both lines are perpendicular to the same line, the transversal, then the two lines will be parallel.0405

OK, let's go over a few examples: Determine which lines are parallel for each.0418

This one right here, the first one, is giving us that angle ABC is congruent to angle (where is D?...) DGF.0426

That means that this angle and this angle are congruent.0440

OK, now again, when we look at angle relationships formed by the transversal, we only need three lines.0445

We have a bunch of lines here; so I want to just try to figure out what three lines I am going to be using for this problem,0456

and ignore the other lines, because they are just there to confuse you.0466

This angle right here is formed from this line and this line, so I know that those two lines, I need.0474

And then, this angle is also formed from this line and this line.0480

So, it will be line CJ, line FN, and line AO; those are my three lines.0484

This line right here--ignore it; this line right here--ignore it.0492

We are only dealing with this line, this line, and this line.0495

And from those three lines, we know that this line, AO, is the transversal, because that is the one that is intersecting the other two lines.0499

So, if this angle and this angle are congruent, what are those angles--what is the angle relationship?0509

They are corresponding; and the postulate that we just went over said that, if corresponding angles are congruent, then the lines are parallel.0517

I can say that line CJ is parallel to line FN.0530

The next one: angle FGO, this angle right here, is congruent to angle NLK.0546

So again, I am using this line, this line, and this line, because it is this angle right here and this angle right here.0557

So, for those two angles, their relationship is alternate exterior angles.0567

If alternate exterior angles are congruent, then the two lines are parallel.0575

And those two lines are going to be, since this FN is a transversal, line AO, parallel to HM.0583

And remember that this is the symbol for "parallel."0598

The measure of angle DBI (where is DBI?), this angle right here, plus the measure of angle BIK0604

(BIK is right here) equals 180, so that these two angles are supplementary.0614

Now, these two angles are consecutive interior angles, or same-side interior angles,0621

which means that if they are supplementary (which they are, because 180 is supplementary), then the two lines are parallel.0629

That is what the theorem says; so I know that line AO, just from this information, is parallel to HM.0637

The theorem says that, if the two angles are supplementary, then the two lines are parallel.0654

OK, the next one: CJ is perpendicular to HK; this is perpendicular, this last one.0661

And then, FN is perpendicular to KN; there is the perpendicular sign.0671

Remember the theorem that said that, if two lines are perpendicular to the same transversal, then the two lines are parallel.0677

So then, from this information, I can say that CJ is parallel to FN.0688

The next example: Find the value of x so that lines l and n will be parallel.0703

I want to make it so that my x-value will make them have some kind of relationship, so that I can use the theorem,0715

saying that I have to make two angles congruent or supplementary--something so that I can conclude that the lines are parallel.0728

Let's see: these two angles right here don't have a relationship; this one is an interior angle, and this one is an exterior angle.0742

But what I can do is use other angle relationships; if I use other angle relationships, then I can find some kind of relationship0754

from the theorems or the postulates that we just went over.0770

What I can do: since this angle and this angle right here are vertical angles, I know that vertical angles are congruent.0774

And since vertical angles are always congruent, and these are vertical angles, since this is 4x + 13,0786

I can say that this is also 4x + 13, because it is vertical, and vertical angles are congruent.0792

Now, this angle and this angle have a special relationship, and that is that they are same-side or consecutive interior angles.0801

I know that, if consecutive interior angles are (not congruent) supplementary, then the lines are parallel.0812

As long as I can prove that these two are supplementary angles, then I can say that the lines are parallel.0823

I am going to make 4x + 13, plus 6x + 7, equal to 180, because again, they are supplementary.0832

Then, I am going to solve for x; this is going to be 4x + 6x is 10x; 13 + 7 is 20; 10x = 160, so x is going to be 16.0851

So, as long as x is 16, then that is going to make these angles supplementary, and then the lines will be parallel.0870

So, x has to be 16 in order for these two lines to be parallel.0880

Find the values of x and y so that opposite sides are parallel.0889

OK, that means that I want AB to be parallel to DC, and I want AD to be parallel to BC.0894

So, the first thing is that...now, this is a little bit hard to see, if you want to think of it as parallel lines and transversals.0902

So, what you can do: if you get a problem like this, you can make these lines a little bit longer.0918

Extend them out, so that they will be easier to see.0924

That means that this one right here and this one right here--if this is the line,0939

and these are the two lines that the transversal is cutting through, then these two angles are going to be consecutive interior angles.0947

This angle and this angle are consecutive interior angles.0956

This angle and this angle are also consecutive interior angles, because it goes line, line, transversal.0959

That means that these are same-side interior angles, or consecutive interior angles.0967

Then, I have options: since I have the option of making this one and this one supplementary, I can also say that this one and this one are supplementary.0973

But this one has the y, and this one has an x; so instead of using x and y to make it supplementary0986

(you are going to have 2 variables), I want to use this one first.0996

I want to solve this way, because it has x, and this has x--the same variable.0999

And you want to stick to the same variable.1005

Here I can say 5x (and I am going to use that, if consecutive interior angles are supplementary, then the lines are parallel) + 9x + 12 = 180.1009

And then, if I do 14x + 12 = 180, 14x =...if I subtract that out, it is going to be 168.1028

Then, x equals, let's see, 12; so if x is 12, then that is my value of x, and then I have to find the value of y.1041

Now, this angle measure, then, if x is 12, will be 60, because 5 times 12 is 60.1063

Then, this right here: 9 times 12, plus 12, is going to be 120.1074

Now, I also know that this and this are supplementary, so the 60 + 120 has to be 180; that is one way to check your answer.1082

Now, remember: earlier, we said that this angle and this angle are also consecutive interior angles, which means that they are supplementary.1092

Well, if this is 60, then this has to be 120, because this angle and this angle are supplementary; they are consecutive interior angles.1099

So, since this is 120, to solve for y, I can just make this whole thing equal to 120.1109

7y + 10 = 120; subtract the 10; I get 7y = 110; and then, y = 110/7.1115

And that is simplified as much as possible, so that would be the answer.1139

Now, when you get a fraction, it is fine; it is OK if you get a fraction.1150

You can leave it as an improper fraction, like that, or you can change it to a mixed number.1153

But this is fine, whichever way you do it, as long as it is simplified1160

(meaning there are no common factors between the top number and the bottom number, the numerator and denominator).1165

Then, you are OK; there is x, and there is y.1173

You solved for the x-value and the y-value, so now that these consecutive interior angles are supplementary, I can say that these sides are parallel.1178

And then, since this one and this one, consecutive interior angles, are supplementary, this side and this side are parallel.1192

Now, notice how, for these, I did it once; and then, for these, I had to do it twice,1202

because any time you have the same number of these little marks, then you are saying that they are congruent;1210

if they are slash marks, then they are congruent; if they are these marks, then they are parallel.1222

If it is one time, then all of the lines with one will be parallel.1226

For these, all of the lines with two will be parallel; if you have another pair of parallel lines, then you can do those three times.1233

OK, the last example: we are going to do a proof.1244

Write a two-column proof: before we begin, we should always look at what is given, what you have to prove, and the diagram.1248

Look at it and see how you are going to get from point A to point B.1260

This is what we are trying to prove: that AB is parallel to EF.1268

Angle 1 and angle 2 are congruent--this is congruent; and then, angle 1 is also congruent to 3.1273

So, one time is congruent...two times are congruent.1299

We know that, since these two angles are congruent, and these two angles are congruent,1310

I know that, since these two angles are going to have some kind of special relationship,1317

my theorem and my postulate say that, if they have a special relationship, then the lines are parallel.1323

So, I am going to just do this step-by-step: here are my statements; my reasons I will put right here.1329

Statement #1: We know that we have to write the given, so it is that (let me write it a little bit higher; I am out of room)...1345

#1 is that angle 1 is congruent to angle 2, and angle 1 is congruent to angle 3.1353

And the reason for this is that it is given.1364

#2: Well, if angle 1 is congruent to angle 2, and angle 1 is congruent to angle 3, then I can say that angle 2 is congruent to angle 3.1370

So, I will read it this way: If angle 2 is congruent to angle 1, and angle 1 is congruent to angle 3, then angle 2 is congruent to angle 3.1387

And this is the transitive property of congruency--not equality, but congruency.1396

#1: If a is equal to b, and b = c, then a = c; that is the transitive property, and we are using congruency, so it is not equality; it is congruency.1412

Then, from there, since I proved that angle 2 is congruent to angle 3, I know that they are alternate interior angles.1427

Alternate interior angles are congruent; that means that I can just say that, since alternate interior angles are congruent, then these lines are parallel.1445

Step 3: You can say that angles 2 and 3 are labeled as alternate interior angles.1456

Or, you can just go ahead and write out what the "prove" statement is--what you are trying to prove,1471

since you already proved that they are congruent, and they are alternate interior angles.1476

Depending on how your teacher wants you to set this up, this will be either step 3 or step 4.1481

My reason is going to be: If alternate interior angles are congruent, then the lines...1491

now, this is not the complete theorem, but you can just shorten it...are parallel.1506

And that would be the proof; and make sure (again, since we haven't done proofs in a while)1517

that the given statement always comes first, and the "prove" statement always comes last.1523

It is like you are trying to get from point A to point B.1526

If you are driving somewhere--you start from your house, and you are driving to school--your house is point A, and your school is point B.1530

There are steps to get there; it is the same thing--proofs are exactly the same way.1541

You need to have your steps to get from point A to point B.1545

That is it for this lesson; we will see you soon; thanks for watching Educator.com.1550

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