Circle Theorems Explained | Easy Geometry Lessons for Students

ello good day and welcome to my channel today I have a question on circle geometry: it says find angle x below now I advise you to pause this video and try to figure out angle x for yourself Figure it out on your own then watch the video all the way to the end and check if you got it right Please let me know what you got in the comment section below—okay back to the video now Let's examine the question: we have a line EAF and this line EAF is tangent to this circle a tangent is a line that touches a circle at exactly one point for example line EAF is a tangent Place this line as a chord within the circle so now if you examine this ad closely forms angle with the tangent line here so this chord AD makes angle w e tangent and if you also examine this chord ad it forms a path a line of a triangle within the circle So look at this triangle ABD the chord AD is a line within the triangle so this AD that forms An angle with the tangent is also a line within this triangle ABD and there is a theorem at we'll use to determine the angle we have right here—let's create a small illustratio Here if you look at this I can make a small triangle here and then draw something like that s now let's call this triangle ABC if this is triangle ABC then angle A here has this s side because here this side because it has this angle similarly this side ac has angle b here and a At the same time this line AB or this side AB has this angle C so I can call this angle C and I'll refer to it as such s is so that if you look at it very closely you'll notice that each line has angle that is opposite to that line so this line has this angle this line has this angle and this line has this angle so with this rule this style and this principle you also argue with me tha This line segment in the triangle inscribed in the circle forms angle of 35 degrees f you look at the chord ad this chord now has this angle of 35 degrees so theorem to apply here is s that the angle which this chord makes with the tangent equals the angle which the same line forms within the triangle it creates circle so this line here now has this angle of 35 and this angle of 35 will also be equal to the e angle that the chord makes with the tangent is 35 degrees and what theorem is applied here The theorem states that angles in alternate segments are equal so this angle here The chord creates alternating segments with this 35 making them equal So I hope that part is clear now moving on let's look at this angle ACE so this triangle ACE f you look at triangle ACE you'll agree that this angle here can be found using the angle in the same segment please note the angle in the same segment is not the same as the angle in the alternate segment The theorem we just applied here is the angle in the alternate segment but one might want to apply the angle in the same segment which are not the same as alternate segments same segment so now if you look at this line AD or let's examine this point A and point D and from this point d you have a line extending to the circumference b and you also have another line from a going to the circumference b and they both form angle of 35 degrees at point b on the circumference from the same point just like the circle so don't forget the key is that they must originate from the same point—so from point a and d we have another pair of lines moving from d to b along the circumference to point c and then from a to c—study it carefully From A to B forms a 35-degree angle similarly from D to C and from A to C—these are the same two lines points we have another line forming angle at the circumference so you agree with me that this The angle will also be 35 just like this one—why Because theorem states that they are angled the same way so these two angles are in the same segment and the key to understanding this is that when they originate from the same point and one extends to the circumference o form angle another set of lines extends to another part of the circumference to create angle Those two angles will be equal because they are in the same segment provided that the pairs of lines are drawn from the same point So this point is 35 and this one will also be 35 Now let's examine this triangle now this complete I'm going to call this angle here angle A and I'll call this entire angle here angle A so if you're looking at these three angles—that is angle A angle 44 and this angle 35—the you will agree that the sum of the three angles in a triangle adds up to 180 so a 44 plus 35 adds up to 180 because the sum of the angles in a triangle equals 180 so This angle along with the 44 and the 35 I have here in triangle ACE add up to 180 Thus we can determine the value of a as being equal to 180 minus 44 minus 35 When these two values are positive and move to the right-hand side they change sign to negative so finally I have a equals 180 minus 44 We have 136 minus 35 which gives us 101 so this complete angle a here is 101 degrees but you agree with me that this full angle a is composed of angle x and angle 35 angle x plus angle 35 equals angle a and angle a equals s 101 so from there we can obtain the value of our x as equal to 101 minus 35 and that will finally give us x is equal to 101 minus 35 which is 66 degrees and that's the value of x so please Don't forget to like this video ask your question in the comment section or if you have any contributions share them there—also please subscribe to this channel if you haven't already Thank you for watching and I'll see you in the next video