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### Measuring Angles in Triangles

- Angle Sum Theorem: The sum of the measure of the angles of a triangle is 180 degrees
- Third Angle Theorem: If two angles of one triangle are congruent to two angles of a second triangle, then the third angles of the triangles are congruent
- Exterior Angle Theorem: The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles
- Triangle Corollaries:
- The acute angles of a right triangle are complementary
- There can be at most one right or obtuse angle in a triangle

### Measuring Angles in Triangles

^{o}, m∠MNO = 75

^{o}, find m∠OMN.

- m∠OMN = 180
^{o}- m∠MON - m∠MNO - m∠OMN = 180
^{o}− 60^{o}− 75^{o}= 45^{o}

^{o}

If two angles of one triangle are congruent to two angles of a second triangle, then the third angles of the triangles are ____ congruent.

^{o}, m∠MON = 45

^{o}, find m∠ONA.

- m∠ONA = m∠OMN + m∠MON

^{o}+ 45

^{o}= 105

^{o}

- m∠ABC + m∠ACB + m∠BAC = 180
- 14 + x + 8 + 2x + 48 = 180
^{o} - 3x + 70 = 180

^{o}, find m∠ACB.

- m∠ACB = 90
^{o}- m∠BAC

^{o}− 40

^{o}= 50

^{o}

There can be a right angle and an obtuse angle in the same triangle.

There must be at least two acute angles in a triangle.

Triangle ABC and triangle DEF, if ∠1 ≅ ∠5, ∠2 ≅ ∠4, then 3 and 8 are ____ congruent.

- m∠ACD = m∠ABC + m∠BAC
- 6x + 9 = 3x + 5 + 2x + 16

- m∠1 + m∠2 + m∠3 = 180
- m∠2 + m∠3 = 180 - m∠1
- 4 ≅ 1
- m∠4 = m∠1
- m∠2 + m∠3 = 180 - m∠4 = 180 - (2x + 3)
- m∠2 + m∠3 = m∠DAE = 5x + 4
- 180 − (2x + 3) = 5x + 4

*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

### Measuring Angles in Triangles

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.

- Intro
- Angle Sum Theorem
- Using Angle Sum Theorem
- Third Angle Theorem
- Exterior Angle Theorem
- Flow Proof of Exterior Angle Theorem
- Triangle Corollaries
- Extra Example 1: Find the Value of x
- Extra Example 2: Find the Value of x
- Extra Example 3: Find the Measure of the Angle
- Extra Example 4: Find the Measure of Each Numbered Angle

- Intro 0:00
- Angle Sum Theorem 0:09
- Angle Sum Theorem for Triangle
- Using Angle Sum Theorem 4:06
- Find the Measure of the Missing Angle
- Third Angle Theorem 4:58
- Example: Third Angle Theorem
- Exterior Angle Theorem 7:58
- Example: Exterior Angle Theorem
- Flow Proof of Exterior Angle Theorem 15:14
- Flow Proof of Exterior Angle Theorem
- Triangle Corollaries 27:21
- Triangle Corollary 1
- Triangle Corollary 2
- 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

### Geometry Online Course

### Transcription: Measuring Angles in Triangles

*Welcome back to Educator.com.*0000

*This next lesson is on angles of triangles, so we are going to be measuring unknown angles within triangles.*0002

*First, the angle sum theorem: this is a very important theorem when it comes to triangles, because you will use it really often.*0013

*And what it says is that the sum of the measures of all of the angles of a triangle adds up to 180.*0026

*So, in this triangle right here, the measure of angle 1, plus the measure of angle 2, plus the measure of angle 3, equals 180.*0033

*Now, this line right here is just to help with this diagram; and that is called an auxiliary line.*0043

*Whenever you draw a line--you just add a line or a line segment to a diagram to help you visualize something--then that is an auxiliary line.*0050

*So, this is an auxiliary line, because it is just to show that the three angles of a triangle will add up to 180.*0062

*Now, if this line right here is parallel to this line, this side, and if I extend this line out right here,*0071

*then from the last chapter we know that, if we have parallel lines, then certain angles have certain relationships.*0085

*And this right here is going to be our transversal; so then, we have two lines that are parallel, and our transversal.*0094

*Parallel line, parallel line, transversal...then angle 3, with this angle right here, are alternate interior angles.*0104

*If we have that, then those angles are alternate interior angles.*0117

*And remember: as long as the lines are parallel, then alternate interior angles are congruent.*0127

*So, I can say that this angle right here and this angle right here are the same--they are congruent--because the lines are parallel.*0133

*Now, the same thing happens here: angle 1 with this angle right here--again, if these two lines are the parallel lines,*0141

*and this is my transversal (so again, it is going like this), then this is one, and this is one.*0151

*They are alternate interior angles; and if the lines are parallel, then they are congruent.*0160

*So, whatever the angle measure is for this, it is going to be the measure of the angle for that.*0170

*Now, looking at this right here: the measure of angle 1, the measure of angle 2, and the measure of angle 3 are going to equal 180.*0177

*Why? Because they form a line; 1, 2, and 3 together form a line.*0187

*We know that a line equals 180; so if it forms 180 this way, then the same thing; angle 1, angle 2, and angle 3 here form 180.*0195

*So, the three angles of a triangle are going to add up to 180; and we know that because of this--*0211

*because, if I look at this angle, the same thing as the measure of angle 1 here, and this angle,*0216

*the same thing as the measure of angle 3 here, all three of them are going to be supplementary,*0223

*which means that these are supplementary--they are the same angles.*0228

*So, the measure of angle 1, plus the measure of angle 2, plus the measure of angle 3, equals 180.*0231

*That is the angle sum theorem: All three angles of a triangle add up to 180.*0237

*So, using the angle sum theorem, let's find the measure of the missing side.*0247

*I am given two angles out of the three; one is missing; I need to find the measure of angle A.*0251

*I can say that x + 35 + 85 is going to add up to 180.*0259

*It is x +...now, if I add this up, it is going to be 120...equals 180; so if I subtract the 120, then I get 60.*0272

*Right here, this is x = 60 degrees.*0292

*The third angle theorem: what we just used right now, that theorem, is called the angle sum theorem,*0300

*where all three angles of a triangle add up to 180.*0307

*This is the third angle theorem, and this is saying that, if you have two triangles, one of these angles is congruent*0311

*to an angle on the other triangle (see how this one and this one are congruent), and a second angle is congruent*0322

*to another angle on the other triangle, then automatically,*0332

*the third angle for this triangle is going to be congruent to the third angle for this triangle.*0338

*So, if two angles of one triangle (which are these two) are congruent to two angles of a second triangle*0345

*(these two), then automatically, the third angles of the triangles are going to be congruent.*0353

*And that is the third angle theorem; and it is just saying that, if I know this angle measure and this angle measure,*0361

*then I have to subtract those two from 180, and I get this angle measure, because all three add up to 180.*0367

*So, if I say that this is, let's say, 80, and this is 60, well, then, those together are going to be 140.*0375

*That means that I have to subtract it from 180, and that is going to give me this angle measure right here.*0390

*Well, this angle and this angle are the same; this angle and this angle are the same;*0395

*again, I have to subtract it from 180; so that means that the sum of these two is going to be the same thing as the sum of those two.*0402

*So, automatically, the third angle has to be the same as this angle here, because it is the same number,*0411

*and I have to subtract the number from 180; so that is the same thing right here,*0416

*and then again...so it is like you are doing the same problem.*0427

*And that is a 4...*0436

*If you have an angle congruent to an angle on the other triangle,*0445

*and a second angle also congruent to another angle on the other triangle,*0448

*then automatically, the third angles will be congruent; and that is the third angle theorem.*0452

*We have the angle sum theorem, and we have the third angle theorem.*0458

*Now, these theorems: again, remember, the theorems are supposed to be proven before we can actually use them.*0462

*But in your book, it will show you, or it will have you do the proof on these theorems; so just take a look at that.*0470

*The exterior angle theorem: now, we know that "exterior angle" sounds familiar.*0480

*An exterior angle is an angle outside; so when it comes to a triangle, what we are dealing with in this lesson here,*0486

*or this chapter, "exterior angle" is referring to an angle outside the triangle.*0493

*So, any time anything is exterior, it is always on the outside.*0501

*So, in this case, it is on the outside of the triangle.*0506

*Now, this is my triangle right here; I have an angle, angle 4; that is my exterior angle.*0510

*Now, I could have more than one exterior angle.*0524

*If I draw this out, then I will have an exterior angle here; if I draw this out, I am going to have an exterior angle here, and so on.*0527

*So, I could have a few different exterior angles, but this is the one that I am looking at right now.*0538

*The measure of an exterior angle (and of course, this theorem applies to any of the exterior angles*0546

*that you see beyond the triangle) is equal to the sum of the measures of the two remote interior angles.*0551

*This is very important to note, because otherwise you don't know what the exterior angle equals.*0567

*There are three angles of a triangle: one of them is adjacent to the exterior angle, which is angle 3.*0581

*So, angle 4 and angle 3 are actually a linear pair.*0592

*Not including that angle, the other two angles that are not next to the exterior angle,*0601

*which are angles 1 and 2, are the remote interior angles.*0610

*Now, if I drew an angle out here, and made this angle 5, the two remote interior angles would be 1 and 3, not 2.*0615

*If I drew an exterior angle out right here, and made this angle 6, then I wouldn't be talking about angle 1; I would be talking about angles 2 and 3.*0625

*The two remote interior angles would be the two angles inside the triangle that are far away from it,*0634

*that are not adjacent, that are not next to the exterior angle.*0641

*That is the remote interior angles; now, think of "remote" as in the TV remote.*0646

*Whenever you use the remote, if you are right next to the TV, you are not going to be using the remote;*0652

*you can touch the buttons on the TV; but the remote is used when you have distance away from the TV.*0657

*In that same way, remote interior angles would be the angles that are some distance away,*0666

*that are not right next to it, but further away--the other two angles.*0672

*The measure of angle 4--that is the exterior angle that we see here in this diagram--*0682

*is going to be equal to the sum of the measures of the two remote interior angles.*0689

*Since we know that that is 1 and 2, this angle measure is going to be the same as the measure of angle 1, plus the measure of angle 2.*0695

*Just to show you the proof of this: the measure of angle 1, plus the measure of angle 2*0717

*(let me do this, actually, in a different color, so you know that this is actually not part of this),*0722

*plus the measure of angle 3--those are the three angles of a triangle--are going to add up to 180; that is the angle sum theorem.*0733

*I know that the measure of angle 4, plus the measure of angle 3, adds up to 180.*0742

*Why?--because it is a linear pair; they make a straight line, so they add up to 180, the measure of angle 4 and angle 3.*0755

*If you look at this, well, the measure of angle 1, plus the measure of angle 2, plus the measure of angle 3, equals 180.*0768

*The measure of angle 4, plus the measure of angle 3, equals 180.*0776

*Doesn't that mean that these two have to equal the same?--because all of this, plus whatever this is, is 180;*0779

*and then, that plus the measure of angle 3, equals 180; so these have to automatically equal each other.*0786

*If I use the substitution property, and I say, "OK, well, then, if all of this equals 180, and all of this equals 180..."*0793

*then I can just make all of this equal to all of this, because they both equal 180.*0801

*So, the measure of angle 1, plus the measure of angle 2, plus the measure of angle 3, equals the measure of angle 4 plus the measure of angle 3.*0807

*And all I did was just...since all of this equals 180, and all of this equals 180, then I can just make these equal each other.*0817

*Then, I just made all of this equal to all of that.*0828

*And that would be the substitution property of equality.*0831

*Then, from here, if I subtract the measure of angle 3 from both, that would be the subtraction property.*0836

*Then, the measure of angle 1, plus the measure of angle 2, is equal to the measure of angle 4.*0849

*That is what this exterior angle theorem is: the measure of angle 1, plus the measure of angle 2, equals the measure of angle 4.*0859

*So, again, just to explain this proof to you: if all of these three angles of a triangle add up to 180,*0866

*and these two add up to 180, well, then, this plus 3 equals 180, and this plus 3 equals 180;*0875

*then these two will equal the measure of angle 4.*0884

*Just take a look at this again, if you are still a little confused.*0893

*Just remember the exterior angle theorem: the measure of angle 4, the exterior angle,*0897

*is equal to the two remote interior angles, the measure of angle 1 and the measure of angle 2.*0901

*OK, so we just went over the proof, explaining the exterior angle theorem.*0916

*But I want to actually go over a flow proof.*0923

*Now, a flow proof is a little different in the format of proofs than what we have done so far.*0929

*We have done two-column proofs, and we have done paragraph proofs.*0940

*A flow proof is another type of setup of a proof; you are just showing it in a different format.*0945

*But it is still the same thing, where you have to show the statement and the reason behind it.*0957

*This one, the proof that we just did, is just a brief proof; it wasn't the actual full step-by-step proof.*0963

*So, we are just going to do the full proof here right now.*0973

*And so, what I am going to do is start with...*0977

*Now, with a flow proof, whenever one step leads to another step, then you are going to draw an arrow.*0982

*It is going to consist of boxes, and each of these boxes is going to flow--*0989

*you are going to draw an arrow to show the next statement and reason, then the next statement and reason,*0999

*and so on and so on, the next steps, until you get to what you are trying to prove.*1003

*Now, remember how we talked about the proofs--how it is from step A to step B.*1010

*If you are driving from point A to point B, then there is a series of steps that you have to take to get there, a series of turns and whatever.*1014

*In the same way, with a proof, you are going to start from point A; and through a series of steps, you are going to arrive at point B.*1025

*That is a proof; this is just another format, so I am just going to show you the flow proof of this.*1035

*The first thing, right here, is going to be: I am going to give you this statement, which is triangle ABC, and that is "Given."*1042

*I am going to write the statement, and then I am going to write the reason below it.*1052

*And then, I am going to draw that as a box.*1056

*And then, like I said, you are going to draw an arrow that leads to the next step, and then to the next step, and so on.*1062

*If it doesn't--let's say you have a step that wasn't from the step before--then you can just write it below here.*1068

*So, I am actually going to do that right now: I am going to say that angle BCD and angle ACB (these two angles) form a linear pair.*1080

*Now, that is not from this step; that is not the next step of this, so I am going to write it below here.*1095

*I am going to say, "Angle BCD and angle ACB form a linear pair."*1102

*And then, my reason is going to be...well, that is just the definition of a linear pair.*1125

*They form a linear pair, so they are a linear pair; this is "definition of linear pair."*1132

*And that is my first box; then, from here, if we form a linear pair, then what does that mean?*1145

*That means that angle BCD and angle ACB are supplementary.*1156

*Whenever two angles form a linear pair, then they are supplementary; and so, my reason behind that is,*1174

*"If two angles form a linear pair, then they are supplementary."*1180

*And I have to write that whole thing out as my reason, because there is no name for it.*1197

*Remember: if there is no name for a theorem or a postulate, then you have to write out the whole thing.*1201

*You can abbreviate stuff, but you have to actually write out the whole sentence.*1205

*That is my second box; then, from here, if they are supplementary, then what?*1212

*I said that they are supplementary; then what is true?*1221

*That means that the measure of angle BCD plus the measure of angle ACB equals 180.*1226

*If they are supplementary, then don't they add up to 180?*1239

*And that is just the definition of supplementary angles.*1242

*Whenever you go from saying that they are supplementary to then making them equal to 180,*1252

*that is the definition of supplementary angles, because that is what the definition says.*1258

*If they are supplementary, then they equal 180.*1262

*Now, see how our flow proof is going this way; it is going to the right.*1269

*A flow proof can also go from top down; you can draw arrows going downwards, too; you can go in that direction.*1275

*OK, now, I want to continue on from this right here.*1285

*Now, look at what I am trying to prove: I am trying to prove that, again, this exterior angle is going to be equal to the measure of the two remote interior angles.*1296

*So, this is the measure of angle BCD; this is equal to the measure of angle A, plus the measure of angle B.*1305

*Like how I showed you before in the last slide, if I have a triangle ABC, then I know that the measure of angle A,*1313

*plus the measure of angle A, plus the measure of angle C, is going to equal 180; what is my reason?--"angle sum theorem."*1323

*We just went over the angle sum theorem.*1338

*Now, in order for you to use the angle sum theorem, you have to have something state that it is a triangle first,*1342

*because the angle sum theorem says "the three angles of a triangle."*1348

*So then, see how this is given: "triangle ABC" is saying the statement that you have a triangle.*1352

*Then, from there, you can say that the three angles of that triangle are going to be 180.*1358

*Again, you have a triangle, and then you can say that the three angles of that triangle are going to add up to 180.*1365

*There is that; and then, since you know that all of this is equal to 180, and all of this is equal to 180--*1373

*this one says that the three angles add up to 180, and this one says that these two add up to 180--*1385

*then, what I can do for my next step is make all of this (since all of this equals 180, and all of this equals 180) equal to all of this,*1393

*since they have the same measure; so I can say that the measure of angle A, plus the measure of angle B,*1407

*plus the measure of angle C, equals the measure of angle BCD, plus the measure of angle ACB.*1420

*See how all of this equals 180, and all of this equals 180; so I just make them equal to each other.*1435

*And what is the reason for that?--"substitution property."*1440

*Now, see how this step right here was derived from this step and this step.*1457

*See how I drew an arrow from that one; I can also draw an arrow from this to that, because both of these led to this one.*1464

*Then, if I look at this right here, let's see: there is one thing that I am going to correct here.*1474

*I don't want to say the measure of angle C here; now, I know this looks like angle C, because it is part of the triangle--*1488

*it is one of the vertices of the triangle; but see how this angle C can refer to three different angles.*1495

*It can refer to this angle right here, this angle right here, or this whole angle right here.*1507

*So, I want to specify this more to "the measure of angle BCA," and the same thing here.*1511

*Or actually, let's change it to ACB, because that is the name that I called it for this one.*1524

*I called this angle ACB, so I will just call this the same angle.*1534

*So then, here also, it is the measure of angle ACB.*1539

*That way, I know, because this one and this one are supposed to be the same,*1546

*because this angle is part of this triangle, and this angle is also part of this.*1551

*So then, the next step...now, I don't have room, so I am going to draw it like that.*1558

*I can say that the measure of...now, I am going to write this part first;*1567

*I am going to use the subtraction property to get rid of this.*1574

*And then, I can say that the measure of angle A, plus the measure of angle B, equals the measure of angle BCD.*1578

*But here in my "prove" statement, I have the measure of BCD first.*1583

*It is the same thing; it is just stating this one first: the measure of angle BCD equals the measure of angle A plus the measure of angle B.*1590

*I just switched it: and the reason for that is the subtraction property of equality.*1601

*And that is it--that is the proof.*1611

*Now, you can do this as a two-column proof; then you would just split up the statements and the reasons, and just number them off.*1615

*You can also write this as a paragraph proof, where you would just write it in sentences; and then, this is the flow proof,*1622

*where you are grouping up the statement with each reason,*1630

*and then you are just drawing the arrows to see how it leads from point A to point B.*1634

*OK, moving on: triangle corollaries: a corollary is kind of like a theorem, where you have to prove it.*1642

*But you can use theorems to prove corollaries; it is not as big of a deal as a theorem, but still necessary.*1654

*The first one is "the acute angles of a right triangle are complementary."*1671

*If I have a right triangle (this is going to be #1, so I am going to write that right here)...this is kind of slippery...*1675

*then since I know that all three angles, this angle plus this angle plus this angle, have to add up to 180;*1695

*but this right here is going to be 90 degrees; well, if this is 90, this right here is just saying that this has to be an acute angle,*1705

*and this has to be an acute angle, because if this is 90, then these two are going to be 90 together.*1722

*Together it has to be 90; and in order for them to be anything but acute, it has to be 90 or greater.*1731

*So, if this angle wants to be not acute, if it wants to be a right or an obtuse angle, then it has to be 90 or greater.*1739

*The same thing here: if this doesn't want to be an acute angle, then it has to be either 90 degrees*1749

*(to be a right angle) or greater than 90 (to be an obtuse angle).*1755

*So, it has no choice but to be acute angles, because they are going to add up to 180.*1758

*And so, since they do add up to 180, if this is, let's say, A, and this is B; then the measure of angle A,*1765

*plus the measure of angle B, is going to equal 90 degrees.*1773

*So then, not only are they acute, but they have to add up to 90, because all together, they are going to add up to 180.*1778

*So, if the measure of angle A, plus (this is C--let's say that that is C) the measure of angle B,*1789

*plus the measure of angle C, is equal to 180--all three angles--well, this right here is equal to 90;*1795

*that means that all of this has to be 90.*1808

*So, if this is 90, then the other two are left to add up to 90.*1818

*This corollary--you don't have to use it, because you can just state it like this.*1825

*But it just makes it easier, because if you have a right triangle, then you don't have to even worry about the right angle.*1833

*You can just say that the other two angles are going to add up to 90.*1839

*The next one, #2: There can be at most one right or obtuse angle in a triangle.*1844

*I kind of already explained #2 when I was explaining #1.*1850

*You can only have one right angle or one obtuse angle in a triangle, again, because if this is 90...*1857

*all three angles have to add up to 180; that means if this is 90, then this plus this plus this has to be 90.*1867

*That way, all three can add up to 180.*1879

*So, let's say that this wants to be a right angle, along with this one.*1883

*Well, then, there is my right angle; that is this one.*1887

*If this wants to be a right angle, too, then it has to go that way, in order for that to be a right angle.*1891

*There is no way that this could be a triangle; so then, you know that there can only be one right angle in a triangle.*1896

*And then again, let's say I have an obtuse angle.*1902

*Let me try to draw more than one obtuse angle in a triangle; there is my obtuse angle there.*1912

*But see, this angle right here wants to be an obtuse angle.*1917

*Then, it has to be greater than 90, so let's say I draw it like that.*1922

*Is there any way that that could be a triangle right there?*1927

*There is no way that this side and this side are going to meet, unless I draw another side, a fourth side.*1932

*And if I have four sides, then that is not a triangle.*1938

*So then, if I have one obtuse angle (this one is greater than 90), then this one has to be less than 90.*1940

*It can't be greater than 90 also, because then this is going to be, let's say, 100, and let's say this is 100.*1949

*That is already 200 degrees right there, and you know that all three angles have to add up to 180.*1957

*So, that is not going to work; that is not going to work.*1964

*At most, there is one right angle, or at most one obtuse angle, in a triangle.*1967

*OK, let's go over our examples: Find the value of x.*1977

*The angle sum theorem: I know that all three angles are going to add up to 180.*1983

*So, 79 + 36 + x = 180; so this is going to be 115 + x = 180; subtract the 115; x is going to be 65.*1988

*The next one: 32 + x + x = 180; 32 +...x + x is...2x = 180; subtract the 32; 2x is going to be 148, and then x is going to be 74.*2022

*So, each one of these is 74.*2051

*Find the value of x, again: here is my exterior angle; the measure of this angle is going to be equal to*2062

*the sum of the two remote interior angles, which are the two angles away from the angle,*2073

*not the one that is next to it (because there are three angles).*2079

*So, the other two: let me just write out the angle measure: 50 + x...those are my two remote interior angles...equals 105.*2082

*Add up these two; it is going to be the same thing as that measure right there.*2100

*So, I subtract the 50, and x will equal 55 degrees.*2103

*Again, the exterior angle is going to be equal to the sum of the two remote interior angles.*2112

*That is 25 + x = 128; I subtract the 25; x = 103, so this angle right here is 103.*2120

*OK, so then, we are going to find the measure of angle A, which is this right here.*2140

*Now, I have a right triangle for this first one; all three angles are going to add up to 180,*2145

*but because I have a right angle, this already uses up 90--it uses up half of it.*2157

*That means that I don't even have to worry about that; I can just say that that means the other two,*2164

*the remaining two angles, are going to add up to the other 90.*2168

*So, if this takes up half already, if this takes up 90 degrees, then these two are going to add up to 90,*2172

*because all three of them together have to add up to 180.*2181

*So, without worrying about that...you could do that; you could just say that this angle,*2183

*plus this angle, plus this angle, is going to add up to 180.*2190

*But I am just going to go ahead and say that these two (forget this angle B, the right angle) are going to add up to 90.*2193

*So, 2x + 15 + 3x - 5 is going to equal 90; again, that is because this one is 90--that means that the other two have to add up to 90,*2202

*because they all have to add up to 180; so this is 5x + 10 = 90; I subtract the 10, so 5x = 80; then x = 16.*2220

*And then, that wouldn't be my answer, because they want us to find the measure of angle A.*2241

*That means that I have to plug x back into angle A.*2247

*So, it would be 2 times 16, plus 15; that is 32 + 15 equals 47, so the measure of angle A is 47.*2252

*And then, the next one: there is my exterior angle; it is equal to the sum of the two remote interior angles.*2273

*So, it would be these two and not this one: 6x - 10 = 45 + 3x + 5.*2281

*So, 6x - 10 =...I can just add these up, so this would be 3x (I can write that better) + 50.*2294

*Then, I am going to subtract 3x, so it is 3x =...add the 10...60, so x = 20.*2310

*And then, here is angle A; so the measure of angle A equals 3(20) + 5; 60 + 5 is 65, so the measure of angle A equals 65 degrees.*2318

*My next example: Find the measure of each numbered angle, which means that I need to find the measure of angle 1,*2341

*the measure of angle 2, and the measure of angle 3.*2347

*Let's see here: I have two triangles; I have a triangle right here, and I have a triangle right here.*2354

*Now, if you get this type of problem--you get a problem that looks a little complicated, a little confusing,*2361

*with kind of a big diagram, then just take a second and just observe it--take a closer look; see what you have.*2370

*I see here that I have two triangles, because we know that this lesson is on triangles.*2380

*I am going to look for triangles; I also have an exterior angle.*2390

*I have two parallel lines; this means that you have parallel lines, right there.*2396

*Now, I want to find all three angles--the measure of all three of them.*2404

*Since I have an exterior angle, I am thinking that I might have to use the exterior angle theorem,*2409

*which I do, because I have two remote interior angles, which is a 42, and then the measure of angle 3.*2415

*To find the measure of angle 3 (the measure of angle 3, plus 42, equals 110--that is the exterior angle theorem--the exterior angle*2424

*equals the sum of the remote interior angles), if I subtract the 42, the measure of angle 3 equals 68.*2439

*Then, if this is 68, let's see here--how am I going to find the measures of angles 1 and 2?*2454

*Well, maybe I can use the triangle sum theorem for 1 and 2.*2470

*But look: I am missing two angles, this angle and this angle; I only have the measure of this angle, so I can't use the angle sum theorem yet.*2477

*But what I can do is look at my parallel lines; if it helps, just extend out the lines so that it will be easier to see your two parallel lines.*2489

*And then, the line that is crossing both is your transversal, so you could extend that, if you want, too.*2501

*And then, you can also move your book, or maybe move your paper, if it is on paper--and try to look at the parallel lines*2508

*so that it would be horizontal or vertical--whichever is easiest for you to see.*2521

*And then, you can see that this angle right here and this angle right here are alternate interior angles.*2527

*Now, 48 and 68--are those alternate interior angles?*2536

*Yes, they are; but because the two lines that are used to form those two angles are not parallel,*2542

*that is why we have two different angle measures--48 here and 68 here.*2553

*If these two lines were parallel, then this one and this one would have to be congruent.*2557

*But those lines are not parallel; so, yes, they are still alternate interior angles;*2564

*but since the lines are not parallel, they are not congruent.*2570

*Their relationship would just be alternate interior angles.*2574

*They are only congruent if the lines are parallel.*2579

*In this case, with these lines and those transversal (because those are the lines that are used to form those angles),*2581

*since the lines are parallel, these two would be congruent; so the measure of angle 2, I know, is going to be 42.*2592

*This one is going to be 42, also.*2603

*Then, how do I find the measure of angle 1?*2607

*Now that I have the two angles of this triangle, I can use the angle sum theorem to find the measure of angle 1.*2613

*So, the measure of angle 1 is going to be added to 48 and 42, and that is going to give you 180 (the angle sum theorem).*2621

*So, the measure of angle 1, plus...this is 90...equals 180; I subtract the 90, so the measure of angle 1 is 90 degrees.*2638

*This one right here is 90 degrees.*2654

*Again, just take a look at your diagram and see what you have; look to see what you don't have and what you can use.*2659

*They are always going to give you what you need, like here: they gave me parallel lines and an exterior angle.*2670

*That is it for this lesson; we will see you next time--thank you for watching Educator.com.*2677

0 answers

Post by Anthony Salamanca on January 13, 2016

In this example, if there was an auxiliary line above vertex which makes two parallel lines then would angle 105 be equal to x? Would this make 105 and x alternate congruent angles?

1 answer

Last reply by: Professor Pyo

Sat Jan 18, 2014 2:55 PM

Post by Yuval Guetta on January 13, 2014

for the flow proof can you write the second statement (that BCD and ACB form a linear pair) before writing the given instead of writing it underneath?

0 answers

Post by Werner Dietrich on May 5, 2011

The class topics/content is located on the left. You can select a topic by scrolling down that list, or you can simply "fast forward" by moving the progress bar below the video.

Hope this helps.

0 answers

Post by Sonia Oglesby on February 23, 2011

How do you fast forward to what you need. I don't need to watch the entire class.