Geometry 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 congruent0311

to an angle on the other triangle (see how this one and this one are congruent), and a second angle is congruent0322

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 triangle0345

(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 angles0546

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 20717

(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 degrees1749

(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 to2062

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 angle2424

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 lines2508

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