[Music] foreign hello and welcome to this video where we're going to learn about Circle theorems before we do our first theorem we're going to need to know the names of some key points of a circle so if we draw a circle and then draw a straight line that goes from one side of the circle to the other this line is known as a chord a chord will always split a circle into two segments here we have a segment at the top and also a segment here at the bottom now let's have a look at our first Circle theorem the first theorem States the angles in the same segment are equal this means if we use this chord that we've drawn to create an angle but keep that angle within the same segment all of those angles will be the same size so let's draw an angle in the upper segment using this chord like this if we draw a second angle like this one or even a third like this one then all of these angles must be the same size because they come from this chord and they're in the same segment so if the purple one was 68 degrees the green one would be as well and so would the blue one sometimes this theorem is drawn without the red chord so we can remove that and it may look something like this and the property still holds it's often drawn with only just two of the angles so if we remove one of them like this we end up with this picture where both of the purple angles here are the same size you can also apply this theorem by drawing a chord here if you do the same thing as we just did but upside down you can show that these two angles are also the same and again we don't need to draw that chord so very often you'll see this theorem drawn like this it's sometimes informally referred to as the bow tie theorem because this looks a little bit like the shape of a bow tie in an exam though you want to avoid that terminology and say that the angles in the same segment are equal but if the bow tie helps you recognize when this is true then that's fine also for the next theorem we're going to need to mark on the center of the circle we're going to draw a chord that goes straight through the center this is known as a diameter if we now do the same thing as we did for the first theorem so draw angles that are in the same segment we know they're going to be equal so if we draw one angle like this a second like this and even a third like this we know all three of these angles must be the same size but it turns out that if the chord that you draw happens to be the diameter of the circle going through the center all of these angles will always be right angles so the second theorem is that the angle in a semicircle is 90 degrees we say this is the angles in a semicircle because that diameter splits the circle in half for the next theorem we're going to draw a chord once again and draw an angle at the circumference also but rather than drawing another angle at the circumference we're now going to draw an angle at the center but using the same chord so something that looks like this if the green angle at the circumference of the circle was 50 degrees then the blue one would be twice this 100 degrees so for this theorem we say the angle at the center is twice the angle at the circumference you could also say in reverse the angle at the circumference is half of the angle at the center once again this theorem still applies if we remove the chord so sometimes you see it drawn like this for the next theorem we need to learn some more terminology if you draw a quadrilateral where all four of the sides touch the circumference of the circle this is known as a cyclic quadrilateral now it's only true if all four of the corners of the shape are on the circumference for example this shape here is not a cyclic quadrilateral because this corner here isn't on the circumference of the circle so let's return to our original one this is a cyclic quadrilateral the theorem states that the opposite angles in a cyclic quadrilateral had to make 180 degrees for example if we look at these two angles here they're opposite each other and they're in a cyclic quadrilateral so if we knew that this one was 86 degrees we'd know the other one must be 94 degrees since 86 plus 94 makes 180. similarly these two angles must add up to 180 as well if we knew this one at the top was 110 the bottom one must be 70. since 110 plus 70 is 180. for the next theorem we're going to draw a tangent to the circle a tangent is a straight line that touches the circle in one place we're then going to draw a radius from the center of the circle to the point where the tangent also touches the circle in this diagram this angle here will always be a right angle so we can say for this theorem a tangent meets a radius at 90 degrees now if we stick with the same diagram we can do some more theorems if we create a point at the end of this tangent on the right hand side and call that P and then if we draw a second tangent from P to the circle it would look something like this if you measure this distance here from P to the point where the tangent touches the circle at the top and the same distance at the bottom these two lines would be the same length we don't normally show this with arrows though we usually draw a line across both of those lines to show they're the same length like this if you are going to write this theorem down you could say that tangents from the same point are equal in length we can also add a bit more information to this diagram if you draw the angle at p then draw a line from the center of the circle to P then this line will split that angle in half or we could say it bisects the angle so if the whole angle here was 30 degrees then we'd have 15 degrees on the top and also 15 degrees on the bottom now we're going to move on to the final theorem if we draw a tangent to the circle and then draw a chord from the point where the tangent touches the circle we form an angle here like this next we're going to use the cord to create an angle at the circumference but not in the same segment in the other segment so something that looks like this it turns out that the angle that the tangent makes with the chord is equal to this angle that the chord makes at the circumference the same happens with the chord on the other side as well so if you look at the chord on the right hand side the angle it makes with the tangent is this green one here but this chord will create an angle at the circumference in the other segment over here so these two angles are also equal we call this the alternate segment theorem and you'll need to know this name here's a quick sketch of all of those theorems we've covered on one page you may find this useful for your revision now we're going to have a look at how we can use these theorems to solve some problems in this question we've been asked to find the angle ABD but we've also been asked to give reasons for our answers so every time we find an angle we need to explain what we did this will often involve just writing down the theorem that we've used so in this one we're looking for the angle ABD we're first of all going to find the angle ACD you should notice that this angle here ACD goes with the angle Ace to make a straight line so these two angles must add to make 180. so we could say that 180 minus 105 equals 75 so we know that this red angle here is 75 degrees and remember we need to give reasons for our answer so we would state that the angles on a straight line add up to 180 degrees now we can use one of our Circle theorems to find the angle ABD this one looks like the bow tie theorem the first one we did so the red angle we just found is actually the same as the angle ABD so this one is also 75 degrees so we could say that angle ABD equals 75 degrees and once again we need to give a reason because the question asked us for one and we would give the first theorem which was the angles in the same segment are equal in this next question we've been asked to find angle BCF but this time we don't need to give reasons because it hasn't asked for them we do need to show working out still though we're going to start by looking at the triangle ACD which is this one here we know this is an isosceles triangle because we can see the a d and DC are the same length this means that the angle ACD down here is 35 degrees it's a good idea to Mark any angles that you find directly onto the diagram but we're also going to write down the angle ACD equals 35 degrees now we're going to stay inside this triangle and work out the angle ADC which is this one here we know angles in a triangle add to make 180 degrees so if we add up the two angles we have 35 at 35 that gives you 70 and then if we subtract this from 180 we get 110. so we can add this to our diagram as well and we're going to write down the angle ADC equals 110 degrees next we're going to look at the whole quadrilateral here you might remember from before that one of the theorems was about cyclic quadrilaterals we can be sure this is a cyclic quadrilateral because a b c and d are all on the circumference the theorem was that the opposite angles in a cyclic quadrilateral make 180 degrees so if we look at the angle that's opposite the 110 that's this one here ABC we can work this one out by subtracting 110 from 180. so if we do 180 take away 110 you get 70 degrees so we can mark this onto the diagram and also write down the angle ABC is 70 degrees next notice how the line a b goes through the center of the circle this means it's the diameter the second theorem we looked at was that the angle in a semicircle is always 90 degrees this means that the angle BCA which is here is a right angle so we can write down that BCA equals 90 degrees now if we stay inside this triangle ABC there's only one more angle to find the one at the top here you can find this by subtracting the other two from 180 so if we do 90 plus 70 that's 160 and then 180 take away 160 is 20 degrees so the angle at the top here is 20 degrees and we can write down the angle c a b is 20 degrees and there's only one more step to go to complete the question we're looking for the angle b c f we can use the alternate segment theorem to show that this is the same as the angle we just found this 20 degrees here is the same as 20 degrees here so the angle BCF equals 20 degrees now we didn't write down any of the worded reasons for this question because it didn't ask us to give reasons for our answers however you can see we have put down quite a substantial amount of working out and we've also labeled every angle that we found onto the diagram it's worth noting that this isn't the only way to solve this question in many Circle theorems questions there are different approaches as long as you get the right answer and show your method clearly you will get full marks to begin this question we're going to look at the line ECF which is a tangent to the circle and also the line OC which is a radius one of the theorems said that a tangent meets a radius at 90 degrees so we know this angle here is 90 degrees so let's write that down angle ocf equals 90 degrees now we're going to look at the whole triangle ocf this one here we have two of the angles in this triangle we have the 32 degrees and the 90 degrees so we can find the missing one if we add together 90 and 32 you get 122 and if you subtract this from 180 you find the missing angle is 58 degrees so we can write the angle C of f equals 58 degrees and label this onto the diagram next we're going to look at the angle c a b which is this one here you should remember that the angle at the center is twice the angle at the circumference we can see the angle at the center from the chord CB is 58 degrees so the angle the chord CB makes up the circumference which is the angle I've marked must be half of 58. so if we do 58 divided by 2 which is 29 we know that this angle must be 29 degrees next we're going to look at the smaller triangle cob this one here sometimes in questions there's information that's not necessarily marked onto the diagram but it is useful it's useful for example to notice that OC and OB are both the radius of the circle this means they're the same length we can mark that onto the diagram but is very often not given to you you need to be able to spot this information yourself this means the triangle cob is an isosceles triangle so two of the angles must be the same so if we subtract 58 from 180 we get 122 and then if you half 122 you get 61. this means both of the base angles in the isosceles triangle must be 61 degrees the one that's important to us is the angle OBC so angle OBC is 61 degrees we can add this to the diagram as well now finally if we look at the triangle ABC we're only one step away from finding the angle which was required in the question so the angle is this one here all of these four angles must make 180 since they're the interior angles of a triangle so if we add up the angles we have 40 plus 61 plus 29 makes 130 and if we subtract this from 180 you get 50 which is the answer to the question and angle above thank you for watching this video I hope you found it useful check out the one I think you should watch next And subscribe so you don't miss out on future videos also check out the exam questions I put in this video's description