Parallelogram Properties

Now let us learn all about parallelograms in this session. Now even though I am using the word all about parallelograms, best thing is it is only this much, that is if you visualize it can get over in a couple of minutes. So if I take a parallelogram like this, where these two are parallel, these two are parallel, if I drop a diagonal here, these two triangles are congruent. If I drop a diagonal here these two triangles are also congruent. Apart from that when I take both the triangles together, these two are congruent, these two triangles are congruent. So you will end up using congruent triangles to prove theorems as well as solve questions. Now there is one more basic concept which all of you are aware of, where which is based on parallel lines and transversals. Now we understand that in a parallelogram, if I take these two parallel lines, this diagonal is a transversal, this side can be taken as a transversal this side can also be taken as a transversal. So the two important basic concepts which you should be well aware of which are which I am sure all of you are, means I am going to revise right now and in the next couple of minutes the criteria on congruency as well as the concept of parallel lines and transversals . And by the time I finish revising it effectively this chapter will also get over. You will be so confident that you can solve any question based on parallelograms using these two very basic concepts and will be able to prove any theorems based on parallelograms and you are going to find it very easy and interesting. Now how to utilize parallel lines and transversal as well as congruent triangle to prove theorems as well as to solve questions based on parallelograms, I will give you a very simple method it is almost like a short cut technique, rather than memorizing, a lot of thermos and practice a lot of questions if you understand this very short discussion, I am sure that all of you will get it but I am going to simplify to a very basic level. So let’s take a parallelogram like this now in this parallelogram, we will look at different scenarios of parallel lines and transversals. So if I take a diagonal like this, so first scenario is where I consider these two parallel lines and this diagonal as a transversal, in that case angle1 will be equal to angle 3 and incase where you have to write the reason, any way you know the reason. Reason is the alternate interior angles where these 2 parallel lines, and this diagonal is transversal. The next scenario can be where these two parallel lines are considered and this same diagonal considered as a transversal, where angle 2 will be equal to angle 4. So angle 2 is equal to angle 4 why? You know the answer. They are alternate interial angles. So these two are very simple scenarios which we will come across and can be used to prove theorems as well as to solve questions. So some of the proves will be based on converse also, see the first one is where when you take these two as parallels , these two equal, now what is the converse? When 1 =3, when first scenario, angle 1= angle3, considering this diagonal as a transversal or the line which is these two you can consider, you can take these two parallel also. That is these two are parallel, these two angles are equal. Or if these two angles are equal, these two lines are parallel. Sometimes we use the converse also to prove theorems. So you keep that in mind. Now the next scenario is , next scenario is when we consider the parallelogram like this in that case consider this two lines are parallel and just this side as transversal . In that case, how are the angle 1 and angle 2 are related? They are the co- interior angles or the interior angles on the same side of the transversal. So what is the result? Angle 1 + angle 2 you can take it as 180 degrees. This method also will be used in some other questions and now if consider these two parallel lines and this as a transversal, and in that case angle 2 and angle 3 as in a diagram, angle 2 + angle 3 will be 180, same reason, interior angles on the same side of transversal. Here also sometimes we will use the converse, what is a converse? If the two angles are adding upto 180, that is angle2 + angle 3 is 180, since they are in the same side of this particular line,you can prove these two lines are parallel. Now these things are very basic method, something which all of you know, but this is how you have to utilize in a parallelogram. Now we will learndeal with intersecting lines also in a parallelogram that is if I take a parallelgram like this and if take both the diagonals, the diagonals will be intersecting, in that case angle 1 will equal to angle 3 and angle 2 will be equal to angle 4 and what is the reason we have to write for this because they are vertically opposite angles, with the intersecting, where the intersecting lines are involved. So angle 1= angle 3. And angle 2 = angle 4 . so this also will be used in questions. Now how congruent triangles are used? So how will you visualize congruent triangles in parallelogram if you ask me it is very simple, take thesane parallelogram and two lines and two lines like this, and same parallelogram if join these two points to get a diagonal, so this triangle ABCD so, this train gle, which is ABC, and this triangle which is CDA, these two are congruent triangles . so will add to a question why that’s what you learnt in using parallel lines and transversals, or the intersecting lines to mark some of the angleto be equal, and some of the sides to be equal, now how to use congruent triangles in parallelogram/ it is very simple, consider a parallelogram ABCD like this in this case if I take a diagonal here, we will get two triangles, these two triangles are congruent. You have to keep that in mind that is triangle ABC and triangle CDA are congruent, so once you get that two triangles are congruent, you have to write the reason, whether it it will only depend on the data given, whether you are using SAS or SSS or SAA it will only depend on the parameters given. Now once you know that triangle ABC is congruent with triangle CDA that is these two triangles, the corresponding parts, that is three pairs of sides to be equal, three pairs of angles to be equal. That is something which all of you know, how will you visualize? These two sides are equal These two sides are equal these two sides are equal because it is a common side, and you also know that these two angles are equal that is 1& 3 and these two angles are equal that is 2&4 and these two angles are equal that is 5&6. The other way around, if I take this diagonal, then also these two triangles will be congruent. It just depend on how you are taking the diagonal. Different ways of getting congruent triangles in parallelogram are what is we are discussing now, now if take the parallelogram back, the same parallelogram, and if consider both the diagonals here, here we will again get congruent triangles, now ABCD the pointi f intersection is o, triangle AOB that is this triangle, is congruent with this triangle, which AOB and COD are congruent. Same way these two triangles are also congruent. Now if we look at the first scenario where AOB congruent with COD, where these two sides will be equal, these two sides will be equal, these two sides will be equal, and these two angles that is 2&4 are equal, and these two angles that is 3&5 are equal, and these two angles that is 1&6 are equal because they are all corresponding parts of congruent triangle. So you will end up using congruent triangles like this also, while proving theorems on parallelograms and solving questions. Thse are very simple methods, once you know how to take congruent triangle in a parallelogram and how you will arrive at the criteria for congruent triangles using parallel lines and transversals, these are almost like shortcut techniques, with minimum practice, you will have the confidence to prove any theorem and solve questions and without memorizing. That is most important part here, because this is you are learning by understanding. And to reach a level where you start analyzing, you can start apply it in questions, I am going to find it very easy and well below your level. So thse are results that we have to use, sometimes using while solving theorems. So these two triangle congruency is an important lessons for proving theorems in parallelograms. It is very simple. You just read or remember this or understand this in diagram there is nothing to memorize. Now you will get the same confidence once you solve a few theorems based on parallelograms, using the same logic or how we are arriving at congruent triangles, and how we are using parallel lines and transversals in a parallelogram. So let’s start proving some of the theorems, in a parallelogram, opposite sides of a parallelogram are equal, opposite angles of parallelogram are equal and each diagonal will divide the parallelogram into two congruent triangles. So these are the three theorems which we will prove together, that is we need to prove that opposite sides are equal, opposite angles are equal and we have to prove also that each diagonal will divide the parallelogram into two congruent triangles. So the very simple proof, we are just going to use what we have discussed before and so will understand that easily. So let’s start with the proof. So let’s take a parallelogram, like this ABCD, now we have to prove that these two sides are equal, these two sides are equal, we need to prove that these two angles are equal, these two angles are equal, and if I take a diagonal like this we also need prove that these two triangles are congruent. We know what we are supposed to prove and we know how to prove. So first thing draw a diagonal, AC, reason you write construction, then second point , angle 1 = angle 3, what is the reason you will write? Parallel lines AB and CD where AC is the transversal, in this case these two angles are alternate interior angles and they are equal. Similarly angle 2 is equal to angle 4, here parallel lines are AD and BC, where the transversal is AC but they are still altrnate interior angles . now AC=AC because it is identity. Next point you can write triangle ABC congruent with triangle CDA, now what will be the reason, you can write the reason is ASA right? because these two angles are equal, these two sides they are common sides so they are equal and the se two angles are equal, that’s why it is ASA. So once after getting triangle ABC, congruent with triangle CDA, now we can easily write corresponding parts of this triangles equal so we can prove. we just proved that this diagonal is bisecting parallelogram into two triangles where they are congruent so the third part is already proved. Now we can directly write down AB=CD and BC= AD, that is from the last statement. We can also write angle A =angle C and angle B= angle D that is opposite angles are equal. So opposite sides are equal proved, and opposite angles equal that is also proved, from the statement triangle ABC congruent with triangle CDA, so it is very simple proof for these three properties. What are the three properties which we just proved? Opposite sides to be equal, opposite angles to be equals, and diagonal is dividing the parallelogram into two triangles, where they are congruent. Very simple and easy. Which is proved that in a parallelogram, opposite sides are equal, opposite angles of are equal, and diagonals, each diagonal divides the parallelogram into two congruent triangles. So it is very easy to prove the converse also. So the next property in a parallelogram, which we are going to prove and simple, very simple one is in a parallelogram, diagonals bisect each other. So to prove this, first take a parallelogram ABCD we need to prove that diagonals AC and BD they bisect each other. That is we need to prove that OD =OB and OA=OC . Now for this first, in triangle AOB and COD, you can write it as first statement, in triangle AOB and COD we have angle 3 = angle 5 , we can write the reason, alternate angles are equal. The parallel sides here are AB and CD and the transversal is AC, and angle 3= angle5, second statement you can write now, angle 1= angle 2, what is the reason? Vertically opposite angles, they are equal. Very simple, and next one we can write, AB=CD because in parallelogram opposite sides are equal. From these three points, we can directly write down triangle AOB and triangle COD are congruent what is the reason? We got the two pairs of the angles to be equal, one pair of sides will be equal, so it si double AAS criteria, AAS criteria is the reason for these two triangles becoming congruent. thsese two triangles are congruent so finally when this theorem is completely based on just you need to understand and based on the patterns we are discussing right? You will have proves as well as questions which can be solved based on these two triangles being congruent, so that is something I have discussed initially in the pattern itself. Since these two triangles are congruent we can directly write down as a result, that OB=OD and OA =OC and that is what we are supposed to prove. Reason is the previous statement. You can write down as a reason. Very simple right? So it is an easy proof. Next we will prove the converse, what is a converse? If the diagonals bisect each other in a quadrilateral, then it is a parallelogram. That is also a very simple proof, we have to use the same two triangles being congruent, just the criteria will change because of the given parameters are different, and what we are supposed to prove is exactly the reverse. So next we will prove that, so to summarize important properties of a parallelogram, we can visualize everything in a single parallelogram, so take a parallelogram like this, opposite sides are equal, opposite angles are equal, and diagonals bisect each other like this. So this is something which you should keep in mind, we will look at some more special parallelograms, as well as some more quadrilaterals. So in a parallelogram, opposite sides are equal, opposite angles are equal. And diagonals bisect each other. So three keen properties of a parallelogram. Now we will understand what are the other properties which will add up as and when we look at other quadrilaterals as well as some special parallelograms.