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### Inscribed Angles

- Inscribed angle: An angle whose vertex is on the circle and whose sides are chords of the circle
- If an angle is inscribed in a circle, then the measure of the angle equals one-half the measure of its intercepted arc
- If two inscribed angles of a circle or congruent circles intercept congruent arcs or the same arc, then the angles are congruent
- If an inscribed angle of a circle interprets a semicircle, then the angle is a right angle
- If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary

### Inscribed Angles

m∠BCF = 30

^{o}, find m⁀BF.

^{o}

write 3 inscribed angles.

m∠BCF = 30

^{o}, find m∠BDF.

^{o},

Determine whether the following statement is true or false.

Points B, C and D are on circle A, ―CD passes through the center A, ∠CBD is a right angle.

If a quadrilateral is inscribed in a circle, then its opposite angles are congruent.

Name a central angle, the inscribed angle, and the intercepted arc.

- Central angle: ∠CAD
- Inscribed angle: ∠CBD
- Intercepted arc : ⁀CD

Inscribed angle: ∠CBD

Intercepted arc : ⁀CD

m∠ACB = 100

^{o}, find m∠BOC.

- m∠ADB + m∠ACB = 180
^{o} - m∠ADB = 80
^{o}

^{o}.

Given: ―AB passes through the center O, ⁀AD ≅⁀AC

Prove: ∆ ABC ≅ ∆ ABD

- Statements; Reasons
- ―AB passes through the center O; Given
- m∠ACB = m∠ADB = 90
^{o}; If an inscribed angle intercepts a semicircle, then the angle is a right angle - ⁀AD ≅ ⁀AC; Given
- ―AD ≅ ―AC; If two arcs of same circle are ≅ ,then the corr. chords are ≅
- ―AB ≅ ―AB ; Reflexive prop. of ( = )
- ∆ ABC ≅ ∆ ABD ; HL

―AB passes through the center O; Given

m∠ACB = m∠ADB = 90

^{o}; If an inscribed angle intercepts a semicircle, then the angle is a right angle

⁀AD ≅ ⁀AC; Given

―AD ≅ ―AC; If two arcs of same circle are ≅ ,then the corr. chords are ≅

―AB ≅ ―AB ; Reflexive prop. of ( = )

∆ ABC ≅ ∆ ABD ; HL

The measure of an inscribed angle is always one half the measure of its intercepted arc.

In a circle, if two inscribed angles have the same intercepted arc, then the two angles are congruent.

*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

### Inscribed Angles

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 0:00
- Inscribed Angles 0:07
- Definition of Inscribed Angles
- Inscribed Angles 0:58
- Inscribed Angle Theorem 1
- Inscribed Angles 3:29
- Inscribed Angle Theorem 2
- Inscribed Angles 4:38
- Inscribed Angle Theorem 3
- Inscribed Quadrilateral 5:50
- Inscribed Quadrilateral
- 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

### Geometry Online Course

### Transcription: 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 thing*0949

*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

1 answer

Last reply by: Briahnna Austin

Wed Apr 20, 2016 2:59 AM

Post by Briahnna Austin on April 20, 2016

Hello, this video was great, but I had a question about the central angles you listed.

By definition you said the central angle is related to the center. The point is in the middle and the chords extend to the end of the circle-- you listed <CPB, and <APB as central angles and I was wondering why is <APC or <CPA not listed as a central angle, since it follows the qualifications a central angle should have?