Geometry Inscribed Angles

Welcome back to Educator.com.0000

For this lesson, we are going to go over inscribed angles of circles.0002

An inscribed angle is an angle within a circle whose vertex is on the circle.0009

We know that the sides of the inscribed angles are chords (remember: chords, again, are segments whose endpoints lie on the circle).0017

So, the vertex and the sides of an inscribed angle are on the circle...and this is the angle, right there.0028

Remember: if this is the angle, then this arc that it is hugging is the intercepted arc.0040

From this point all the way to here--this arc is intercepted by this inscribed angle.0049

The inscribed angle is half the measure of the intercepted arc.0060

Now, we went over central angles; we know that central angles are angles whose vertex is on the center.0069

Be careful with central angles and inscribed angles.0077

The arc with the central angle are congruent; we know that the intercepted arc and the central angle have the same measure.0083

But the intercepted arc with the inscribed angle is different; the arc has double the measure of the inscribed angle.0092

If this arc measure, let's say (the measure of arc AB), is θ, then the measure of angle ACB is θ divided by 2.0102

This angle is half the measure of this arc; the measure of angle ACB is half the measure of the arc.0126

You just take the arc measure, and you divide it by 2 to get the inscribed angle.0147

Be careful: the biggest mistake with this, the most common mistake, would be: if the arc, let's say, is 100,0152

then this angle is 50; be careful that this is the bigger measure.0163

I have seen...if this is 100, then sometimes students make that mistake and make this 200--multiply it by 2,0173

thinking that this arc is half the measure of the angle; don't get that confused.0180

Make sure that the intercepted arc is double the angle, or that the angle is half the arc.0184

If this is θ, then the arc will be 2θ; if the arc is θ, then this would be 1/2θ.0195

Always just remember that the arc is bigger than the inscribed angle.0204

Now, if we have two inscribed angles with the same intercepted arc--here we have this black angle right here--0211

that is the inscribed angle; the intercepted arc is from here to here; and we have another inscribed angle, the red angle,0219

using the same intercepted arc; well, let's say that this has a measure of 100; if the arc has a measure of 100,0229

then this inscribed angle is 50 degrees; then this inscribed angle is also 50 degrees,0239

because it is 1/2 the measure of the intercepted arc, and they are both intercepting the same arc.0249

So then, these have to be congruent; all inscribed angles with the same intercepted arc are congruent.0255

Again, they are both inscribed angles with the same intercepted arc.0267

That means that these angles have to be congruent.0273

OK, if an inscribed angle of a circle intercepts a semicircle, then the angle is a right angle.0281

We know that, let's say, if you draw a diameter right there, that makes this a semicircle.0289

This semicircle has a measure of 180; this is the inscribed angle with this intercepted arc.0299

So, the inscribed angle intercepts a semicircle; since we know that the inscribed angle is half the measure of the intercepted arc,0311

well, if the intercepted arc is 180, then the inscribed angle has to have a measure of 90.0322

Remember: the inscribed angle is half, so if this is 180, then this has to be 90, which makes this a right angle.0328

Again, a semicircle is 180; half of that is 90, which makes that the measure of the inscribed angle; therefore, it is a right angle.0339

An inscribed quadrilateral: now, we know that an inscribed polygon is when a polygon is inside a circle,0353

with all of the vertices touching a circle; so here, we have a quadrilateral...0361

Now, this is not a rectangle; we know that it is nothing special; it is just a quadrilateral, inscribed.0366

If any type of quadrilateral is inscribed in a circle, opposite angles (angle B with angle D are opposite angles) are supplementary.0372

And angle C with angle A are going to be supplementary.0386

The measure of angle B, plus the measure of angle D, is going to equal 180.0392

The measure of angle C with the measure of angle A is also going to be 180.0400

We know that all four angles of a quadrilateral always add up to 360.0409

So then, these two, B and D, will add up to 180; and C and A will add up to 180.0414

Our examples: the first one: Name the central angle, the inscribed angle, and the intercepted arc.0424

Here, there are a couple of central angles here; but they are asking for the central angle and the inscribed angle that share the same intercepted arc.0434

Now, it doesn't matter; you can just name any central angle.0448

Here, I know that the central angle is when the angle is inside a circle with the vertex at the center ("central"--think of "center").0451

This one and this one are always confused a lot, so be careful; with central, the vertex is on the center;0467

an inscribed angle is the angle where the vertex is on the circle.0476

Central angle and intercepted arc have the same measure; the inscribed angle is half the measure of the intercepted arc.0482

Here is the central angle; the central angle is angle CPB.0492

I can also say angle APB--that is another one.0505

An inscribed angle is, again, an angle whose vertex is on the circle; that would be right there, angle CAB.0513

And then, the intercepted arc for each one: for the central angle CPB, the intercepted arc would be CB; that is the arc.0536

The intercepted arc for this one is arc AB; and then, for this one, it is going to be arc CB.0548

Find the value of x: the first one: we have a circle with an inscribed angle, and that is x; that is what we are looking for.0566

This side is a semicircle; this side is, also; so this intercepted arc with this arc is 92 degrees.0586

Since I know that the inscribed angle is half the measure of the intercepted arc, as long as I can find the measure of this arc, I can find x.0596

I am going to take 180 (because a semicircle is 180: this is 180, but I don't have to worry about that), minus the 92; then I will get this arc.0607

So, once I find this arc, then I can just divide that by 2 and get the inscribed angle.0621

180 - 92 is going to be 88; if this is 88, then what is x?0628

x is 88 times 2, or divided by 2? Well, remember again: the angle is smaller than the arc, so the inscribed angle is half.0645

So, it would be x = 88/2, which is 44; so that right there, that inscribed angle, has a measure of 44 degrees.0658

And the next one: they give us three angles of the inscribed quadrilateral in a circle.0675

I know that opposite (now, be careful here; it is not consecutive--it is opposite) angles of an inscribed quadrilateral are supplementary.0689

If I want to find x, then I have to use this angle and this angle here.0703

Be careful that you don't do 98 + this angle; again, it is not consecutive angles that are supplementary.0708

3x - 6 + x + 18 is going to be 180; so here, I have 4x + 12 is going to equal 180; 4x...if I subtract 12, I am going to get 168;0715

and then, if I divide the 4, then I am going to get 42; so x is 42.0749

Now, all they are asking for is the value of x, so that would be the answer.0759

But if they were asking us for the actual angle measure, then we would have to plug this number back in and solve for the angle.0764

This is not the angle measure; this is just x; so you would take the 42 and plug it back into this;0771

and for this angle measure, you are going to get 42 + 18 (I will just do that in red); 42 + 18 is going to give us 60 degrees; this one is 60.0778

And then, this one here: I can just subtract it from 180, because again, supplementary is what we used to find x.0796

You can say that this is going to be 180 - 60, which is 120; or if you want to just double-check your answer,0807

and just solve it out, even though you know it is going to be 120; just plug in x of 42 - 6.0814

3 times 42 is 126, minus the 6 is 120; so we know that that is correct.0827

For the next example, we are going to find the measure of each of these.0841

Here, I have the measure of arc DC, 60 degrees; I have, let's see, chords; I have a diameter; I have a radius; I have inscribed angles.0848

Here is an inscribed angle; here is an inscribed angle...central angles.0869

The first one: the measure of angle CPD--now, since this is the center, this angle, we know, is called a central angle,0877

because the vertex is on the center of the circle; so, central angles have the same measure as the intercepted arc.0891

If the intercepted arc is DC, and it has the measure of 60, then the measure of angle CPD has to also be 60 degrees.0905

The next one: the measure of arc BAC--it is this major arc here; we know that it is a major arc, because it gives us BAC, not just BC.0918

If they said BC, then it would be this arc right here, BC, the minor arc; but BAC is all the way around here.0932

And then, to find the measure of that...well, I have this little piece right here; do I have this and this?0939

Well, I don't what the measure of arc BA is; I don't know what the measure of arc AD is; but I know that together, this whole thing0949

is going to be 180, because it is a semicircle; so 180 + 60 is going to give us the measure of arc BAC;0962

so this is going to be 180 + 60, which is going to be 240.0974

And the last one, the measure of angle BAC: BAC is this angle; it is BAC, so this angle right here is what they are looking for.0987

We know that that is an inscribed angle; if that is an inscribed angle, the intercepted arc would be arc BC;1003

all of this right here is the intercepted arc for this angle here; so do we know the measure of the intercepted arc?1015

Well, they didn't tell us, but we can find it, because BCD, that arc, is a semicircle.1024

So, if this is 60, then this has to be 100 minus the 60; so this will be 120.1034

Then, the central angle here, BPC, is going to also be 120.1045

And then, what about this angle BAC--is it 2 times the intercepted arc, or half the intercepted arc?1052

We know that the inscribed angle is half the arc; so if this is 120, then the measure of angle BAC has to be 120/2, which is 60 degrees.1060

And the last example is going to be a proof: here is the center--the center is at P; what is given?1080

Arc BC is congruent to arc AD; so this is congruent to this; and prove that triangle BCP, this triangle here, is congruent to triangle ADP.1091

I am going to do a two-column proof with my statements and my reasons.1116

Now, remember that, since we are trying to prove that two triangles are congruent...1130

remember the unit where we had to use either Side-Side-Side, Side-Angle-Side, Angle-Side-Angle, or Angle-Angle-Side.1137

We have to use one of these four theorems and postulate to prove that triangles are congruent.1152

That means one corresponding side, another corresponding side, and then a third corresponding side.1159

Those parts have to be congruent, and then we can prove that those triangles are congruent.1166

The first statement is going to be arc BC being congruent to arc AD; the reason for that is "given."1174

And the next step: if those arcs are congruent, then I know that these chords are congruent.1190

And these chords are parts of the triangles; so I am going to say that BC is congruent to AD.1203

And then, I am making them segments of the triangle; what is my reason?1218

It is the theorem that says that, if two arcs of the same circle are congruent, then corresponding chords are congruent.1225

I am just giving my reason there: this theorem that says that, if two arcs from the same circle (or from congruent circles)1256

are the same--are congruent--then their corresponding chords are congruent--that theorem doesn't have a name.1269

So, you just have to write it out; you can shorten it--I shortened it a little bit.1278

And then, the third one: that gives us a side, and they are all using sides; now, what else can I say?1284

I can say (here is a side; I am just going to put S there, so that way I know that I proved that one of the sides is congruent),1296

from these triangles, that these angles are congruent; so angle BPC is congruent to angle APD.1308

And my reason for that is "vertical angles are congruent"; any time that you have vertical angles, they are always congruent.1330

So then, there is an angle there; and then what else?--I need one more.1343

I have a side, and I have an angle; that means that I am not going to be using this one.1351

And for the next one, if these chords were parallel, then I could say that this angle is congruent to this angle, because they are alternate interior angles.1357

But I don't know that they are parallel, so I can't use that reason.1375

Now, what I can say, though, is that this angle here, angle B, is an inscribed angle.1383

I will just draw this so that it is easier to see; this is the inscribed angle, right here, with an intercepted arc CD.1395

This angle right here is an inscribed angle intercepting this arc; now, this angle here, angle A, is also an inscribed angle,1406

so this angle and this angle are both inscribed angles, intercepting the same arc.1423

What do we know about two inscribed angles with the same intercepted arc? They are congruent.1435

Just to make it easier to see, here is angle B, and here is angle A; this is the same, and this is the same, so they are congruent.1442

This one and this one...angle CBP is congruent to angle DAP (I wrote B), and the reason for that: "Inscribed..."1469

and again, this theorem doesn't have a name, so you just have to explain it "...angles with the same intercepted arc are congruent."1503

So, here is another angle; now, since we have two angles, and we have a side, this can either be this one or this one.1531

We have to see what the order is: is the side the included side from those?1544

We have Angle-Angle-Side; it wouldn't be Angle-Side-Angle, so it would be this one right here that we are using.1553

My fifth and final statement is going to be the statement right here: Triangle BCP is congruent to triangle ADP.1565

What is the reason for that? Angle-Angle-Side.1582

To review, the given was that this arc and this arc are congruent; since they are within the same circle,1594

their corresponding chords will be congruent; and that is what we used for the first one right here; that is a side.1602

Then, for the third one, we said that this angle and this angle are congruent, because they are vertical angles; and that is an angle, right there.1611

And then, we said that this angle B and angle A are congruent, because they are both inscribed angles intersecting the same arc.1622

It is like if this is, let's say, 80 degrees, the inscribed angle is half the intercepted arc; so if this is 80, then this has to be 40.1634

Well, this is also an inscribed angle with that same arc; so if this is 80, then this has to be 40.1647

So then, inscribed angles with the same intercepted arc are congruent, and that was the last piece that we needed to prove that the triangles are congruent.1653

And the rule is Angle-Angle-Side; that is it for this example.1661

And that is it for this lesson; thank you for watching Educator.com.1670