The Triangle & Its Properties - Chapter 6 - Introduction - Class 7| Insightful maths
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00:00Hello everyone. Welcome to Insightful Maths. In today's session, we are going to discuss
00:07the chapter Triangle and its properties, which is grade 7 chapter. It's a one-shot video
00:12in which we are going to discuss all the topics which are there in this chapter. So sit along
00:18with a rough notebook and a pencil so that you can take the notes and you can do the
00:22questions along with me. So let's begin the session and don't forget to see the video
00:27till end. Alright, let's begin. So all these topics that we are going to cover in this
00:33chapter are, this is what is a median, altitude, the sum of the length of two sides of a triangle
00:40that means triangle inequality theorem, exterior angle, third side property, Pythagoras theorem
00:47or the questions on Pythagorean triplet and the properties of isosceles and equilateral
00:52triangle. All these things we are going to discuss in this chapter. Okay, so the first
00:58thing now we need to understand is what is a median and an altitude. You can have a look
01:04at this picture. This is the vertex of the triangle. We have triangle PQR. It has three
01:11vertices P, Q and R. If I draw a perpendicular, perpendicular means if these are, if let's
01:21say these are two lines and they are crossing each other at 90 degree angle, they are known
01:28as perpendicular lines. For example, these two, these two are perpendicular. That means
01:33they make 90 degree angle with each other. So from a particular vertex, if I draw a perpendicular
01:41on the base, on the opposite side, you have a vertex P. What is its opposite side? Angle
01:49P is formed by this line segment QP and PR. Which line segment is not used for making
01:56P? That is QR. So this line segment QR, this side, this is opposite to angle P. So when
02:03we put perpendicular to P, this perpendicular or the height is known as the altitude. So
02:10what is the median? Median is again a line segment, which if drawn from the vertex to
02:19the middle point of the opposite side. For this vertex P, what is the opposite side?
02:25QR. If at its middle point, at M, I join this line segment PM, that is the median. You can
02:33remember like this, M stands for middle point. See again, if I draw a triangle like this,
02:41this is A, B and C. If I want to draw a median from B, what will I do? First I will see what
02:49is the opposite side of B. You see, B is formed by this side and this side. So which side
02:56is not used and which is the opposite side? This one. So we will take the midpoint of
03:03this opposite side, let's say B and I join B with D. I will join B and D, that is the
03:11median. Similarly, if I have to draw the median from point A, we will go to the middle point
03:18of the opposite side. Likewise, from C, we will go to the middle point of the opposite
03:23side. So median always goes from a vertex to the middle point of the opposite side.
03:30You see, how many medians are possible in a triangle? It is 3. Maximum 3 medians are
03:36possible. All the medians are inside the triangle. None of the medians can be outside. So what
03:43can be outside? Altitude of a triangle may be drawn outside also. Now one more interesting
03:51thing is, I told you that altitude is the height of the triangle. If I draw a right
03:57angle triangle, please listen to this very carefully. If this is let's say a right angle
04:03triangle, I asked you, what is the height and what is the altitude? So if this is A,
04:09B and C, this side AB, is it the height of the triangle? This is the height, right? So
04:17this side AB, this is the altitude and this is the side of your triangle. So in the case
04:24of a right angle triangle, whether I take this side or this side, both these sides serve
04:31as an altitude. Now why did I call this BC as altitude? If I flip this triangle, flip
04:38as in BC like this. Let me draw it once again. BC is vertical and AB base is like this. Something
04:48like this. I can place it like this. Either it can rest on AB or BC. Now we are making
04:54it rest on AB. A and B. This can also be the base, AB. Then BC top and this is AC. Same
05:02figure. AC hypotenuse, hypotenuse. This side which was BC earlier, now it has become the
05:08altitude. So in the case of a right angle triangle, both these sides which are making
05:1590 degree angle, both the sides are your altitude. They are giving you the height. Now I told
05:23you that altitude can be drawn outside the triangle as well. Let's see how. Triangle
05:29means a three sided polygon. It is not necessary that the triangle should always be like this.
05:36Like this. This is also a triangle, right? Let me name the vertices A, B and C. If I
05:43want to put median from A, I will go to the opposite side. You see, this line segment
05:49will come inside. If I want to put median from C, I will put median from A. So median
05:54will remain inside. If I want to draw altitude from A, what is altitude? It is perpendicular.
06:01If I draw a perpendicular from A, I have to extend this base. Extend it. You see, this
06:11is altitude. Perpendicular, it is lying outside the triangle. So you will remember, altitude
06:18again, three altitudes and three medians are possible. Medians are always inside the triangle.
06:25Altitude may be outside the triangle as well. A quick recap. Median is a line segment which
06:33joins a vertex with the middle point of the opposite side. Altitude is the height or the
06:41perpendicular drawn from a vertex to the opposite side. Just the perpendicular height. It is
06:48not necessary to pass from the center. Okay, so this is done. Let us go to the next slide.
06:55Again, very important theorem. Generally, in your examination, question comes from this
07:00theorem. Triangle inequality theorem says that in any triangle, the sum of two sides
07:08of a triangle is always greater than the third side. For example, if you take any two sides
07:16of a triangle, the sum of both sides should always be greater than the third side. If
07:21it is not so, whether the sum is lesser or equal to, that triangle is not at all possible.
07:29So if I, you know, the question comes, can 2, 3, 5, for example, if I want to draw a
07:37triangle, given it 2 cm, 3 cm and 5 cm. Question is, can these be the sides of any triangle?
07:45There is a difference. Can these be the sides of any triangle? If I was asked about right
07:53angle triangle, I would have checked something else. In any triangle, you will use inequality
07:58theorem. What does it say? You have to check all three inequalities. It is not that you
08:05check true or answer is yes. You have to check all three. Only if it is greater than
08:10all three, the answer is yes, otherwise no. So what we will do? One by one, we have to
08:14make three combinations. 2 plus, let's say first we take 2 plus 5. We took 2 first, then
08:22took 5. Added both, 2 plus 5 is 7. Which side is left when I took 2 and 5? Third side is
08:293. Is this 7 greater than 3? This is fine. First one is true. Now I took 2 and 5. Another
08:37combination I am taking 3 and 5. 3 plus 5. What does that equal to? 8. When I took 3
08:44and 5, what is left on the third side? 2. 8 is greater than 2. This is also fine. Now
08:51next combination is 2 and 3. Now if I add 2 plus 3, what is my answer? 5. And third
08:59side is 5. Is it greater than 5? It is equal to 5. Inequality says the sum of two sides
09:06should always be greater than the third side. Here it is equal. It is a cross here. So your
09:11final answer will be no such triangle is possible. And in the bracket, you will be mentioning
09:18using the triangle inequality theorem. When you check this, you have to write the name
09:25of the triangle. Otherwise marks get deducted. Let's do one more question on this. Let's
09:31say the sides are 2, 3 and 4. I am taking smaller numbers for you to be calculated easier
09:39and understand the topic. Take the combination 2 and 3. 2 plus 3. It is equal to 5. Third
09:47side is 4. 5 is greater than 4. This is fine. Now I am taking the combination 2 and 4. 2
09:54plus 4. That is 6. When we took 2 and 4, 3 was left. It is greater than 3. This is also
10:02fine. We took 2, 3, 2, 4. Now 3 and 4 are left. 3 plus 4. That is 7. When we took 3,
10:094, 2 is left out. The third side is greater than 2. In all three inequalities, the sum
10:15of two sides is greater than the third side. The answer is yes. This triangle is possible
10:23using triangle inequality theorem. In any question, when you are given three sides and
10:31you are asked whether this side can be of any triangle, what will you do? You will add
10:37two sides and check if it is coming greater than the third side. You have to do this for
10:42every combination. You have to show three inequalities. It is not true in one and second.
10:49For each and every combination, it has to be true. Done? I am writing a question here.
10:54You can do this later on. In the comment option, you can write your answer if this is true
10:59or not. I am giving you the sides 6, 7 and 8. You can check and tell me in the comment
11:07if any such triangle is possible or not. Let's move to the next topic now.
11:13Angle sum property is one of the easiest and undoubtedly the most interesting and important
11:18property. It says that in any triangle, the sum of the interior angles in any triangle
11:26be it right angle triangle, equilateral or scalene, the sum of these three interior angles
11:36will always be equal to 180 degree. That means the property, angle sum property of a triangle
11:43says that the sum of the interior angles in any triangle is always equal to 180 degrees.
11:51If I say, let us do one or two questions so that you remember this property well. I have
12:00made this triangle. I am giving you two angles and you need to find the third one. A, B and
12:07C. I have made a box here. I am not giving you the value. This one is 60. You need to
12:14find the value of x. Angles are always measured in degrees. Ensure to put a degree symbol
12:33at the top and write this angle sum property in the bracket. Whenever any box is being
12:41made like this, it always means that these two lines are perpendicular and the angle
12:47between them is 90 degrees. If it is not written like this, it means it is 90 degrees. Sum
12:55of all three is 180. So we will write 90 plus 60 plus x that is equal to 180. In the bracket,
13:03you will write the name of the property, angle sum property. Now 90 plus 60 that is 150.
13:13So that means 150 plus x it is 180. If I transpose 150 on another side, plus becomes minus. So
13:24x is equal to 180 minus 150. The difference of both is 30 along with this degree symbol
13:32answer is 30 degrees. Let us do one more question. Take some interest in the questions. You will
13:44understand it yourself. The more questions you do, the more confidence you will gain.
13:51That's my question. I have given you this angle as 60 degrees. Angle C, mujhe nahi pata.
13:58Yeh wala angle, let's say this is 70. Aapko E ki value nikaliye. The value of x. Ab hum
14:05kya karein? Can we tell what is the value of this interior angle? Yes. You must have
14:14studied that in your lines and angles. Any two lines which are making a cross like this,
14:22this angle and this angle, they are equal. Similarly, this angle and this angle, they
14:27are again equal. They are known as vertically opposite angles. Agar aapko yeh nahi aata
14:35hai, I have already made a video on lines and angles. Please go and watch that video.
14:40It will be much clearer for you. So there is a cross being made here. So this angle
14:45at the top and this angle at the down should be equal. Pehle aap isko mention karogi this
14:51is 60 degree and what is the reason? Vertically opposite angles. Kitna clear hogaya? Aap niche
14:58dekho mere paas ek triangle hai C, D and E. I said in any triangle, the sum of three interior
15:05angles is 180. Toh in teeno ka sum 180 hona chahiye. Pehle aap equation frame karoge 60,
15:11yeh wala. Another angle 70, another angle x. In teeno ka sum 180, bracket mein aap property
15:18angle sum property. Now 70 plus 60 is how much? 130. 130 plus x is 180. 130 ko transpose
15:29karo. So x is the difference of 180 and 130. Answer is 50 degrees. That's your answer.
15:37It's an easy property. I hope you are able to understand this one. Correct? So you can
15:42try the questions based on this property and let me know if you are able to understand
15:46this. Another one very interesting property is the exterior angle property. Let me first
15:54explain you what is an exterior angle. Triangle dekho hum sabko pata hai. Agar yeh A, B aur
16:00C hai. Hain na? Yeh angle hai. This one is let's say A, this is angle B and this is angle
16:08C. Thik hai? Agar main is triangle ki kisi bhi ek side ko extend kar deti hoon. Dekho
16:13main isko extend kar diya. Kisi bhi ek side ko agar aap extend karte hoon, yeh wala extended
16:21side, triangle ki saath ek angle banata hai. And this angle is outside the triangle. Isi
16:28liye hum isko bolte hain exterior angle. Thik hai? Yeh jo exterior angle hota hai na, this
16:35is the sum of interior opposite, two interior opposite angles. Humne bola do angle, do
16:43no interior hona chahiye, dono opposite hona chahiye. Let us apply some common sense. Do
16:48not get confused kaunse do angle ko add karein. Seeda sa pehle ek cheez dekho. On a straight
16:55line, these two angles, the sum of these two angles is 180 aur hum isse kya bolte hain?
17:02Linear pair. Right? The angles on a straight line, sum is 180, they are known as linear
17:07pairs. Ab yaha par, is this angle D making a linear pair with C? Aap dekho bilkoo straight
17:14line banata hai. Toh exterior angle ke saath, jo interior angle joined hai already. Dekho
17:21yeh already joined hai. Wo opposite kaise ho sakta hai? Wo toh isko help par raha hai
17:26linear angle hai na, linear pair banane mein. Humne kya bola? Opposite interior angles.
17:30Toh yeh iske opposite toh nahi hai, yeh toh directly connected hai. Toh jo interior angle
17:36exterior se directly connected hai. Isko aap chhod do. Other two angles you have to
17:41take. So that means, this exterior angle D is the sum of A and B. Samajh aaya? Let me
17:50extend another side, then I think you will be able to understand it better. Ab maine
17:55yeh side extend kiya, this angle I am taking it E. The exterior angle E hai, yeh bhi andar
18:04ke saath do angle ka sum hona chahiye. Ab hume toh nahi pata kaunse do angle ka sum
18:09hai. Maine aapko kya bataya? E ke saath directly kaunsa interior angle connected hai? B. Aap
18:16B ko nahi loge. So E is the sum of A and C. Eksterior angle is the sum of two interior
18:25opposite angles. Iss se related ek question karte hain. This is my triangle. It is like
18:34this. I will just name it. A, B, C and D. Hume yeh wala angle X nikalna hai. It is given
18:45yeh wala jo angle hai 60 hai aur yeh wala angle bhi 60 hai. So what is the value of
18:52X? Pehli baat aise questions mein jo bhi aap property use kar rahe hain, you have to name
18:57that property exterior angle property. Ab aap dekho yeh exterior angle angle C se directly
19:04joined hai na, toh iska matlab hume yeh wala angle toh lena hi nahi hai. So X should be
19:09equal to the sum of these two. 60 plus 60. Toh value kitna aaya? X is equal to 120 degrees.
19:20Ab hum isme kya karte hain? Let us make some minor changes. Ab the question is, hume eksterior
19:28angle is given. Ab mujhe bahar wala angle pata hai aur mujhe andar wala ek angle is missing.
19:34Let's do this. Hume randomly safe values ko add nahi karna hai. Agar eksterior angle
19:40nikalna hai, tabhi in dono ko add karenge. In few questions, aapko interior aur exterior
19:46given hota hai ek interior missing. Let me explain. Agar mein bolte hoon yeh wala angle
19:51is 130 degrees. Yeh 130 degree hai. Aapka yeh wala jo angle hai, let's say this is 70
19:59degrees aur hume X ki value chahiye. 130 is directly connected to C. Toh hume 130
20:08aur C linear pair hai na, iska matlab yeh wala angle se hume matlab nahi hai.
20:13Eksterior angle 130, it is equal to the sum of, aapko 130 plus 70 nahi karna hai, yeh 200
20:21nahi hai. Eksterior angle is the sum of these two. 70 plus X. Toh X ki value kitna hua?
20:3170 ko aap yaha lehao. That is 130 minus 70 is X. Answer is 60 degrees. Toh seedhi si
20:39baat kya hai? Agar exterior angle nikalna hai, toh aap in dono ko add karoge. Agar interior
20:45angle missing hai, toh exterior minus is interior. Theek hai? I hope this is understood. So please
20:53do your textbook questions related to this. You will be able to do that easily.
20:58Next topic pe aa jaao, third side rule. Very easy and understood. Very much basic it is.
21:04Only one last question is there in one of the exercises related to this rule.
21:09Agar kisi bhi triangle ki aapko do side pata hai, so third side ka range hota hai.
21:15Range means, let's understand, mere paas do side hai triangle ki 6 and 7 cm each.
21:22Question says, third side of this triangle lies between which two numbers?
21:29Yaani third side sabse chota kitna ho sakta hai, sabse bada kitna ho sakta hai. Uska na
21:34ek range hota hai. Toh this formula you need to remember. Aapko in dono ko add karna hai,
21:407 plus 6 yaha par. 7 plus 6 is 13. And what is the difference of these two? 7 minus 6
21:49is 1. Aapka ye jo third side hai na, ye dono ke sum se chota hona chahiye, dono ke difference
21:56se bada hona chahiye, equal nahi chahiye. Aap khud bana ke dekh sakte ho, agar ek side
22:017 hai, ek side 6 hai, aur aapne third side 1 liya, aap wo triangle bana hi nahi paoge,
22:06it will not be closed. Similarly, if one side is 7 and another is 6, aap teesra 13 lete
22:12ho, you won't be able to make that triangle. Theek hai, toh seedhi si cheezein aap yaad
22:17rakho, agar aapse third side poocha jaye, ki do side given hai, third side minimum
22:22kitna ho sakta hai aur maximum kitna ho sakta hai. Aap ye formula yaad rakho. It is greater
22:27than the difference, it is lesser than the sum. So if I ask you the question, the two
22:32sides of a triangle are 2cm and 5cm each. What is the range of the third side? Toh aam
22:40kya karenge, in dono ka difference kitna hai? Bigger minus smaller, 5 minus 2, 3. Dono
22:47ka is 7. Aapka jo third side hai, 7 se chota, 3 se bada. Theek hai, that's your answer.
22:55Let's move to the next topic now. I have made a separate video on Pythagoras theorem.
23:02It is the most important topic of this chapter. Agar aapne wo video abhi tak nahi dekha hai,
23:08please go and watch that video. One question from Pythagoras theorem is a must in any of
23:14the examinations. May it be your half yearly examinations of grade 7 or may be it is a
23:19final examination. Ek question aata hi aata hai because this is one of the most important
23:24theorems in this chapter. Okay, so I am not repeating that up. For that you need to see
23:30that video. I have solved few questions also there. You will get the clarity. I am just
23:35giving you a small hint or a recap what does it says. In any right angled triangle, this
23:42theorem holds true only in the case of a right angled triangle. A, B and C. This side, the
23:52longest side is known as the hypotenuse. This is the base and this is the altitude. Pythagoras
24:06theorem says hypotenuse ka square, longest side square is equal to the sum of kiska sum
24:15in dono side ka alag alag square ka sum. A B square plus B C square. In teeno me se do
24:24cheez pata hai, ek missing hai to we can easily find that out. Humne ek triangle inequality
24:29theorem kiya tha na, jahaan pe aapko three numbers given the. Aur aapse poocha tha ki
24:34aaye number koi triangle bana sakte hain. To humne kya kiya tha, we have added two numbers.
24:40It should be greater than the third side. That is triangle inequality. Same question
24:45agar aapko poocha jata hai. There is a difference. Can these be the sides of a right angled
24:52triangle? Jahaan pe bhi right angled triangle bola jaayega, aapko dhyan mein rakhna hai
24:59right angled triangle mein kya apply hota hai Pythagoras theorem. To aapko yeh teen
25:04side given hai aur aapse poocha jaayega, are they making a Pythagorean triplet? Triplet
25:09ka matlab hota hai group of three. Pythagorean triplet means are they following the Pythagoras
25:15theorem? Hai na? Pythagorean triplet agar yeh hai, to yeh Pythagoras theorem ko follow
25:20karna chahiye. Pythagoras theorem says longest side ka square, hypotenuse ka square. Yaha
25:26par given nahi hoga hypotenuse kya hai. You have to take it yourself. 2, 3 and 4 mein.
25:32Greatest is 4. So agar 4 ka square, you have to write, if 4 square is equal to 2 square
25:40plus 3 square, then in the bracket following Pythagoras theorem, this will be a triplet.
25:464 square kitna hota hai? 16. 2 square is 4 plus 3 square is 9. What is 9 plus 4? It
25:55is 13. Left hand side is 16 but they are not equal. Yeh to equal nahi hai. So this 2, 3,
26:024 is not a Pythagorean triplet. Answer is not. Theek hai? Yeh Pythagorean triplet nahi
26:09hai. I am taking another example. Let's say 2, 3 and 5. Ab aapse poocha jaayega, is it
26:16a Pythagorean triplet? Aur Pythagorean triplet naam nahi diya, yeh poocha ja sakta hai, can
26:22these be the side of a right angle triangle? Right angle ka naam jaha bhi aaya, aap kya
26:27use karoge? Pythagoras theorem. Take the longest side. 5 ka square should be equal to 2 square
26:35plus 3 square. Hai na? 5 square matlab 5 multiplied by 5. That is 25. 2 ka square
26:43matlab 2 multiplied by 2. It is 4. 3 ka square is 3 into 3, which is 9. 9 plus 4? It is
26:5213. Left hand side is 25. Yeh ho sakta hai Pythagorean triplet? Answer is again no because
27:00these two sides are not equal. Theek hai? So any such question where it is asked, can
27:06these be the sides of a right angle triangle or Pythagorean triplet? Aap isme yeh hi check
27:12karoge. I am giving you a question here. Quickly check and put in the comment whether these
27:17are the sides of a right angle triangle or not. Theek hai? I have changed the sequence.
27:31I have given you a triplet 5, 3 and 4. Aapko check karke pataana hai, can these be the
27:37sides of a right angle triangle or not? I will wait for your answers in the comment.
27:41Next, now this is the last topic of this chapter. Aap dekho kitni asaani se chapter pura ho gaya.
27:47Agar aapko yeh sab aa gaya na, please do your NCERT questions. Okay? You will be able to
27:53do those easily. Agar nahi hoga, put that in the comment option. I will make a video
27:58for solving those questions as well. Now, properties of some special triangles.
28:04Hum do special triangle padhenge is chapter mein. That is isosceles and equilateral.
28:08Isosceles triangle kya hota hai? Jisme koi bhi do side equal hoti hain.
28:14Equilateral mein saadi side equal hoti hain. So what is special about equilateral?
28:19This is a triangle where all the three sides are equal and all the three angles are equal.
28:26Theek hai? Har ek angle equal hai 60 degree.
28:30Ek aur interesting cheez hoti hai isme. Yeh dekho, humne altitude draw kiya
28:34aur jo yaha pe point aaya hai na, let's say this is D, yeh B aur C ka midpoint bhi hai.
28:41So in the case of an equilateral triangle, median and altitude they overlap
28:47or they coincide each other. Ek ke upar ek hi aata hai, alag alag nahi hota.
28:51Jaise hum koi basic aisa triangle banate hai na, toh yaha se median toh midpoint tak jayega,
28:57altitude perpendicular hai, toh dono alag alag hai na, lekin agar aapke paas
29:01ek equilateral triangle hai, yeh midpoint hai.
29:04Median bhi yahi hai aur altitude bhi yahi hai, koi difference nahi hota.
29:08Theek hai? Come to the isosceles triangle now.
29:12Aapko pata hai, doh side equal hai, same thing jo humne equilateral mein padha,
29:17iska bhi na, median aur altitude overlap karte hai.
29:21Yeh midpoint hai iska, let's say D, toh yeh altitude bhi hai, yeh median bhi hai,
29:26dono mein same hai. Another category or another interesting
29:29property for isosceles triangle is, yeh jo doh equal side hai na,
29:34yeh base ke saath equal angle banati hai.
29:38So you will be getting the questions in your exercise related to this property.
29:43Equal sides make equal angle with the base in the case of an isosceles triangle.
29:49Ab iska kya matlab hai? Aap ek cheez samjho, aapko main ek isosceles triangle banati hoon.
29:55Let me just change the shape.
29:57Theek hai? Aap iske samjh paoge.
30:00Kaunsi doh side equal hai? Let's say yeh dono side equal hai.
30:04I am just naming it up, A, B and C.
30:07Mene aapko bola, in the case of an isosceles triangle,
30:11equal sides make equal angle with the base.
30:15Ab mujhe samjhe nahi aana, kaunse doh angle equal honge.
30:18Asaan sa tariqa kya hai?
30:20Yeh jo doh equal side, jis point pe cut karti hai,
30:23yeh dekho, yeh aur yeh equal hai.
30:26Kis point pe yeh mil rahi hai? B pe.
30:28Aap is point ko chhod do.
30:30Jis point pe yeh intersect kar rahi hai na, usko chhod do.
30:33Other two angles will be considered as the base angles and they will be equal.
30:38Right? Aap yaha pe dekho, yeh wali side aur yeh wali side equal thi na.
30:43Agar hum extent pe rahi hai, yeh kahan pe mil rahi hai?
30:46Is point pe, hum isko chhod denge.
30:48These two angles are equal.
30:50Right? I will just make one more figure.
30:53Just check it out, if you are able to understand.
31:03Alright.
31:06Okay.
31:08So if I say, this is an isosceles triangle.
31:11A, B and C.
31:14Agar main aapko bolte hu, yeh dono side equal hai.
31:18Kaunse doh angle equal honge?
31:20Yeh dono side kahan pe cut kar rahi hai?
31:21Aap A ko chhod do.
31:23Then these two base angles will be equal.
31:26Theek hai.
31:28Doosra aap main figure banati hoon.
31:30Let's say, this and this.
31:33And I join this up.
31:35This is let's say A, B and C.
31:38Ab yeh doh side equal hai.
31:40Toh kaunse doh angle equal honge?
31:42Aap dekho, this AC and BC.
31:46They are meeting each other at which vertex?
31:49Take your time.
31:51Another one minute.
31:53And decide what should be your answer.
31:55This triangle I have given to you.
31:57And I have written here AC and BC.
32:00Yeh doh side hai AC and BC.
32:02Yeh dono equal hai.
32:04Yeh kahan par mil rahi hai?
32:06C par mil rahi hai.
32:08Aap C ko chhod do.
32:10So which two angles will be equal?
32:12A and B?
32:14Theek hai.
32:16Understood?
32:17Good to go.
32:19So please do the questions based on this property.
32:21It will make your life much easier.
32:23And you can do your questions easily.
32:26So please do the NCERT questions.
32:28See you in the next session.
32:30And please don't forget to like and subscribe the channel.
32:33Thank you so much.