Angles in a polygon ExamSolutions

hello everyone and welcome to this tutorial on angles in polygon now in this tutorial we'll run through some important facts that you need to know so you can tackle exam questions successfully so let's have look at the definition polygon well polygon is multi-sided 2d shape made of straight lines all of these are examples of polygons because you can see they're 2d shapes and they're made of straight lines now let's work out the angles inside our polygons and to do this we're going to use the basic angle fact that there's 180 degrees in triangle now knowing this we can apply it to all our polygons by simply splitting them into triangles so for this quadrilateral you can see it's made of two triangles so the sum of angles is 360 degrees for our pentagon you can see the sum of angles is made of three triangles so it's 540 degrees the sum of angles for our hexagon is made up of one two three four triangles which gives me sum of angles to be 720 degrees now an easy way to work out the sum of angles is using this formula where the sum of angles in any polygon is 180 degrees multiplied by minus two where represents the number of sides for example looking at our hexagon this is made of six sides so six subtract two is four four times 180 gives my 720 so this is quick and easy way to work out the sum of angles in any polygon so now we know the sum of angles in any polygon let's have look at some key words regular and irregular well well regular polygon is where all the lengths are equal and all the angles are equal an irregular polygon is where all the lengths are different and all the angles are different now let's have look at some more keywords in particular exterior angles well all i'm going to do is just draw this polygon i'm going to highlight the exterior angles to do this all need to do is elongate length elongate another length and elongate another length the angle created outside of this polygon are called exterior angles so here i've highlighted all my exterior angles now what's really important to note about exterior angles is when bring them all together so all i'm going to do is highlight the center of my polygon and i'm going to bring together all my exterior angles and what you'll notice is the sum of the exterior angles will always be 360 degrees so even if look at irregular polygon here you can see i've extended each of our lengths and then highlighted each of our exterior angles now i'm going to bring together all my exterior angles into the center of my polygon and you can see the sum of the exterior angles is 360 degrees therefore this is the next formula that's super important to know the sum of exterior angles is always 360 degrees now let's extend this fact little bit more using our knowledge on regular polygons well if we know we have regular polygon here and we know all the lengths are equal and all the angles are equal and we also know all the exterior angles sum to 360 degrees therefore we can quickly work out the exterior angle of any regular polygon and it's simply 360 divided by the number of sides this will always give us the exterior angle of any regular polygon this is the key information needed to tackle any exam question regarding angles in polygon so let's have look at past exam question here the question states that to to is straight line to to is straight line and to to is straight line we're asked to work out angle and we must give reasons for our answer so let's start by identifying what we know well first of all we know to to is straight line so therefore can work out angle well is simply 180 degrees subtract our 70 degrees because we know the sum of angles on straight line is 180 degrees therefore we know to be 110 degrees so now we know the exterior angles we have 135 110 and and we know the sum of exterior angles is 360 degrees so that means we can work out which is simply 360. subtract 110 subtract 135 which gives us angle to be 115 degrees because we know the sum of exterior angles is always 360 degrees now the great thing about mats is this is always more than one way to get your solution you could have worked out the interior angles of our polygon and then calculated angle from there either way it still would have confirmed to be 115 degrees now let's have look at another question here the question says the diagram shows regular octagon and regular hexagon we're asked to find the size of angle marked and we must show all working out now remember how we find an exterior angle it's simply extending the length of our polygon so extending the length of our polygon you can see that is made up of the exterior angle of our regular octagon and the exterior angle of our regular hexagon so let's work out the exterior angle of our regular octagon well to work out the exterior angle of regular octagon it's simply 360 divided by 8 which is 45 degrees so now know the angle here is 45 degrees to work out the exterior angle of regular hexagon it's simply 360 divided by 6 which means my exterior angle is 60 degrees so that means is simply the sum of 45 and 60 which is 105 degrees like said before the great thing about maps is there's more than one way to get your solution you could have worked out the interior angles of our polygons and then worked out from there either way you still would have got to be 105 degrees so now let's have look at another exam question here it states that and is regular octagon we know is straight line and we know to to is exactly the same angle as to to now we have to show that the angle is 135 degrees see if you can give it go using the key information that we've looked at and press pause if you need so firstly let's identify the sum of all the angles in our octagon and then from here we can identify what each angle is well to work out the sum of angles of an octagon it's simply 180 multiplied by the eight subtract two so it's 180 times 6 which is 1080 degrees so therefore we know the sum of all the angles in our octagon is 1080 degrees but just want to find one angle so therefore i'm simply going to divide by 1080 by 8 giving me 135 degrees so therefore know each angle inside my octagon has got to be 135 degrees now looking at angle to to know it's going to be the same as to to so can identify these two angles to be 135 degrees now look at the quadrilateral have here well know the sum of angles in quadrilateral is 360 degrees and also know angle bad is the same as cda so therefore can work angle cda to be 360 subtract 135 subtract 135 and divide by 2. this gives me angle cda to be 45 degrees because the sum of angles in our quadrilateral is 360. now can easily work out angle cdj because angles on straight line sum to 180 degrees so therefore know cdj is 135 degrees which confirms what the question told me the great thing in maths is there's more than one way to find the solution so don't worry if you found the answer to be 135 but in different way as long as your working out is clear so now let's have look at our last question it says and are two pentagons of the same size we're asked to work out angle to to and we must show how we got our answer now it's quite difficult to see these regular pentagons so i'm going to simply highlight here's my first regular pentagon and here's my second regular pentagon now from here given the fact that these are regular pentagons it's fair to say that is rhombus because we know these sides are equal in length identifying our rhombus is super important for what we're going to do later on see if we can use this hint as well as this key information to work out angle to to and press pause if you need so remember we know and is rhombus now let's work out each angle inside our regular pentagon and to do this well let's work out the sum of angles of our pentagon first well we know our pentagon has 5 sides 5 subtract 2 is 3 and three times or 180 is 540 so we know all the angles in our regular pentagon add up to 540. so let's find out what each angle is well 540 divided by 5 is 108 so inside my rhombus i'm going to identify my angles to be 108 and also abc to be 108 this is because we know opposite angles in rhombus are the same now we're able to work out angle because this is the same as hcb well we know it's 360 subtract 108 subtract 108 divided by 2 would give us angle this means is 72 degrees so now know is 72 degrees can work out angle to to remember this is our inside angle of our regular pentagon and know the sum of the angle of the regular pentagon is 108 degrees so therefore simply do 108 take away my 72 stating that my angle to to is 36 degrees so in summary when tackling exam questions using ankles and polygon it's super important to remember these key words as well as these key formulas if you like this video please give us thumbs up leave your comments down below and subscribe to this channel so you'll be the first to know when we release our next videos