Introduction to Quadrilaterals
This session is about quadrilaterals, especially parallelograms. Now before we start discussing about parallelograms, by this time, I am assuming that by this time, all of you know- if we take two points and if we join them, we get a line. And now, if we take three points, which are non-collinear and if we join them together like this like this like this- we get a triangle. What happens when there are four points and if we join them in pairs, we will see next? When all four are colinear, like this, one-two-three-four, we get a straight line. When three out of the four are collinear like this two three four on the same line- colinear means same line. When three out of four are colinear, we get a triangle like this, if we join them in pairs. And what will happen when no three are colinear? We will show you the scenario, so it’s very easy to understand. When no three points are on the same line, that is like this, one two three four, here if we join them in pairs, what we get is nothing but a quadrilateral. This is a quadrilateral, now quadric means four and ladrial means sides and that means basically it will have four sides. So will discuss more about these kinds of diagrams, very simple diagrams, in this particular session. Now a simpler way of looking at this, in common language, we start with a line, two lines together you get an angle, with three of them together you get a triangle. Now, two triangles together, if we take them like this, we get a quadrilateral. So a quadrilateral is in a way there are two triangles put together. So, that is why if you ask me in one triangle sum of the all the angles are 180, if you take two of them together, it is three sixty. So, when a quadrilateral, the angels will add up to three sixty, we will anyway prove it in the usual format, the way we usually supposed to do it in exams. But in common language it is very easy to understand why in a quadrilateral the angles will add up to 360. Now, let’s take a quadrilateral a, b, c, d like this, and now this will have four sides- ab, bc, cd, da. It will have four angles- angle a , angle b, angle c and angle d. and it will have four vertices which is a b c and d. lets know more about different types of quadrilaterals and their properties especially parallelograms in this session. So let’s start with the property of a quadrilateral which is true for all quadrilaterals. Its Angle Sum property, now what is Angle Sum property? In a quadrilateral, four angles will add up to 360 degrees. Now the proof for this basic theorem, it’s a very basic proof, take a quadrilateral a b c d like this, now take a diagonal ac. Now why are we taking this diagonal here? The reason is so that you are dividing the quadrilateral into two triangles. And using the Angle Sum property of the triangle only we need to prove the Angle Sum property of a quadrilateral. And here as you can see, I have marked the angles. Now in triangles a d c- one plus two plus two equals to 180 degree what is the reason? You can write the reason in the bracket- Angle Sum property of the triangle. So some of the angles in the triangle will be 180 degrees. And the same way, in triangle abc, triangle three plus five plus six equal to 18 degrees. Reason is the same reason. You can write the reason when you are writing the exam. So write the first one as first, and second one you mark it as second equation. Now what will happen if you take one plus two? Very simple- if you consider one plus two, angle two plus angle three is angle a, angle four plus angle five is angle c, angle one you replace it by d and angle 6 you replace it by b. and you simply take angle one plus two plus three plus four plus five plus six equal to 360 that is one plus two. And in next you replace two plus three by a, four plus five by c, and angle one by d and angle six by d, you will simply get angle a plus angle b plus angle c plus angle d equal to 360. Thus it proves that some of the angles in a quadrilateral is 360. It’s a very simple proof here right? And what’s the logic here? By dropping the diagonal you are converting the quadrilateral to two triagonals. That’s just how you understand a quadrilateral. It’s nothing but two triangles put together, so we can use Angle Sum property of a triangle to prove the Angle Sum property of a quadrilateral. It’s very simple, there is nothing to memorize, you can write the answer in four simple steps. And why some of the angles in a quadrilateral is 360? And let’s understand the Angle Sum property of a quadrilateral by visualization. And since this visualization this method is going to be very simple. So take a quadrilateral like this- mark the angles as one two there four. Now take three more copies, rotate the first piece so that angle two and one are now sitting together. Rotate the second piece so that two one four are now kept together. Rotate the third piece even further so that two one four three are kept together to form a perfect circle, adding up to angle 360. So Angle Sum property is proved. So, you just visualized it, it’s so intuitive and so simple. and you can visualize the Angle Sum property of a quadrilateral in a much simpler fashion like this. Take a quadrilateral, drop the diagonal so that you get it divide it into two triangles. So in a triangle, angles will add up to 180, so two triangles will add up to 360. Now to understand why angles will add up to 180 in a triangle, you can visualize Angle Sum property of a triangle like this. To prove Angle Sum property, that’s in simple language angles in a triangle adding up to 180, I am going to show you a method, the easiest ever and I am sure you haven’t seen it before. So take a triangle name the angles as one two three, join the two big points here, drop to perpendiculars- fold this angle, fold this angle, fold this angle like this, as you can see here one two three are on the same straight line. So one plus two plus three is 180. Insanely simple right? You try this using a paper, it’s simple and thrilling too. It’s almost like wow! Learning like this is so interesting and you can try proving this in multiple ways and the moment you start proving that, that’s when you really improve the way you think. You can actually extend the same logic of dividing the polygon into angles to find the sum of angles of other polygons also. So, to find the pattern , let’s start with a triangle. So, in a triangle like this, there are three sides. The angles are 180 degrees. Now in a quadrilateral like this, if I am dividing it using this diagonal, there are two triangles. So quadrilateral with four sides, and then there are four sides, there will be two triangles. So, what is that is the pattern here? There is difference of two, four minus two. Since there are two triangles Angle Sum will be two into 180 that is 360. So this pattern what you should observe, when there are three sides, it’s a single triangle. When there are four sides, its two triangles. Now let’s see what will happen when there are five sides. Lets take a pentagon like this. And if I divide it using these diagonals there are one two three triangles. So when there are five sides, there are three triangles, so sum of angles will be 3×180 that is 540. Very simple, right? It’s a clear pattern, now, one more if I take a hexagon like this, and if I used these diagonals you can see one two three four triangles. That means when there are six sides four triangles. It’s a clear pattern, right. Six sides four triangles. Sum of angles will be 4×180 seven twenty. So what is the pattern? Three sides one triangle. Four sides two triangle. Five sides three triangles. Six sides four triangles. Its just a difference of two right? So, if you want to generalize if there are n sides there will be n-2 triangles. If you want to find out the sum of all the angles of a polygon with ten sides, as an example, how will you find out? Its easy. There are ten sides. There will be 10-2 eight triangles. So into 180 that is (10-2)X180 that is 8×180 that is sum of all the angles will be 1440.so it’s a clear pattern an interesting pattern using which you can find out some of the internal angles of a polygon. In an n-sided polygon there will be n-2 triangles. you want to find the sum of angles, you will multiply it with 180. Very easy right? Now, if you want to find out the exterior angles of a polygon. sum of the exterior angles of any polygon will be 360 degrees. We will understand it through a very simple activity. Lets draw a polygon, draw a polygon on the ground so if I just show that here if I take a pentagon a b c d e. now start at point a. move along line ab walk along this line ab, after reaching point b, you will be taking a turn equal to angle one and move along bc and on reaching c , turn by an angle equal to 2 on reaching d turn by an angle 3 move along d, on reaching e you will turn an equal to 4 and on reaching the a, you will turn by an angle equal to five and if you do this much and you are in the original position on line ab, by the time you would have made one complete turn which is equivalent of one complete turn is nothing but equivalent of 360 degrees. So its very easy to understand using this activity. So of any polygon sum of exterior angle will be 360. And as discussed before sum of interior angles will be depend on if there are n sides, there will be n-2 triangles and (n-2)x180 you will get the sum of interior angles. And sum of the exterior angles of any polygon is 360 degrees. Let’s look at these basic examples here. In this particular question, so sum of the so here x+90+50+110= 360 so the unknown here which is x you can easily find out which is 110. You will get 110 which is 360-250. So in this question sum of all exterior angles equal to 360 we know that. Each angle is equal to 45. So number of exterior angles will be 360 by 40 which is equal to 8, that means polygon has eight sides. Because you know that the sum of all exterior angles any polygon is 360. Here it’s given that each angles is 45 degree. So just by dividing 360 by 45 you get 8 and that is the number of sides.