So in this class, I want us to consider still working with the properties of our lines, working with the lines that we had. There is the geometry of the lines, but now considering some other properties to be involved, which is the properties of triangles and also the quadrilaterals that we are going to have. So before we consider these typical cases that we are given, let us just have some properties that we can take into consideration. Properties of a triangle.
Let us consider a triangle. What is it that we refer to? We had this triangle from our grade seven.
As we understand, we're working with the types of triangle, all types of triangles that you can be given. But in this part of the geometry of the straight lines, the major part that we are going to consider on a triangle is one as we refer to any triangle this is any triangle any whether it is going to be an acute angle triangle is going to be obtuse angle triangle but any triangle that you refer about the angles that are inside they add up to 180 degrees the angles that are inside of the triangle add up to 180 degrees meaning to say this angle uh to add it to this angle added to this angle adds up to 180 degrees so you're simply saying Angles in a triangle. So that is angles in a triangle adds up to 180 degrees. As long you are in any triangle.
So that is the major part that you are going to have. Before we talk about having an isosceles triangle to say, what are the properties of an isosceles triangle? What are the properties of an equilateral triangle? before we talk about that do you understand the triangle on its own so it's there's a major concept there angles they end up to 180 all right we also have the types of the triangles uh that might be important when you are considering not to say an acute angled or an and just angled by the type like you're working.
with a scalene it's not important but you just need to understand that maybe you are working with a scalene triangle what are the properties that we are going to consider what is it that we are going to consider the sides angles they are not the same so this is the type of a triangle where you are that's a normal triangle that you can have where the sides and angles are not the same all right sorry for that i think it's my network this side but saying here if you are given a scalene the sides these sides that we are seeing they are not equal meaning to say angles also are not equal so in this case sides and angles are not equal so this one i'm just putting it um it's not that important for us but you just need to understand uh like what i was explaining just need to understand the basics then also we consider an isosceles triangle so we also have uh an isosceles triangle all right so this is short for triangle triangle All right. So we also have an isosceles triangle. This one, we have a condition which is very, very important where two sides are equal. If these two sides are equal, like this side and this side, they are equal. The angles that are opposite to this side.
OK, consider this. That's a side. The angle opposite is this one. That's a side.
OK. The angle opposite is this one. So it follows that these angles that I indicated, this angle and this angle, they are also equal. All right. So we are referring to a triangle that is two equal sides.
OK, the two sides are equal. So it can be it can be given in different ways. We can have our triangle like this. Then they put the lines here.
What does it mean? If this side and this side are equal, you take the opposite one. This one goes here. This one, it goes this side.
So it means these two angles are equal. This one and this one. Okay?
So you consider two sides being equal. Two sides are equal. So you consider also the angles, these ones.
that are also equal. So you can even just call them two base angles, that is base angles are equal also. So that's another part that you need to understand on an isosceles.
Then we consider the last one, which is an equilateral triangle. So from the word equi, equal, equal. So we are simply talking about a triangle whose sides are equal all the sides this side this side this side all the sides the angles also they are equal so all sides are equal all sides are equal so in this case each angle here inside is equal to 60 degrees why remember we said angles in a triangle they add up to 100 180 degrees.
But these three angles that we are seeing, they are equal. So it's going to be 180 degrees divided by three angles that are equal. You are going to obtain 60 degrees.
So it means each angle that is there is going to be 60 degrees. This one is going to be 60 degrees. This one is going to be 60 degrees.
Whenever you see a triangle like this, you have these interior angles, the angles that are inside. of 60, 60, 60 like that. It is an equilateral triangle. So these properties, they are going to help us here and there in answering our questions with the parallel lines.
So we shall see how are we going to include these properties. We also talk of the quadrilaterals. Quadrilaterals.
All right. These are now shapes that we consider with four sides. So we consider shapes with what? Four sides. So as long as a shape has got four sides.
So there are some special quadrilaterals that you need to understand. All right. We all know guys about the square.
Or we don't know the basics of a square where all sides are equal. So you consider all sides equal. We know. about our rectangle.
Okay, we know about our rectangle where two opposite sides are equal from our grade 7. We do understand this. So I'm going to add from these ones that we are used to, there are some quadrilaterals which are taken from these basic quadrilaterals that we are used to. Alright, let's consider one, a parallelogram.
That's another quadrilateral that you need to understand, a parallelogram. All right. What is it that we consider as a parallelogram?
It is two pairs of opposite sides parallel. That's the major part. So we are considering something of this nature.
It can be of this shape. Just like what we have on a rectangle, but it is now being shifted a little bit. Like that part of a transformation, it has been transformed.
So it's simply a rectangle that has been transformed through shear. But that is not important for now. What is important is that... A parallelogram has got sides which are parallel, these two sides, opposite sides. They are not just parallel, they are parallel and equal at the same time.
So these two sides, parallel and equal, also consider these two sides that are opposite, these two sides. They are parallel to each other and also they are equal. So we consider that to be the major property.
So we've got two pairs of opposite. opposite sides are parallel, parallel and also equal in length. So we have to consider the ones that are opposite.
They'll be parallel and equal in what? In length. OK, so meaning to say these two, they are parallel and equal in length. So if they are parallel and equal in length, meaning to say.
We can consider the properties of what? The parallel lines. If these two lines are parallel, it means the angles that are inside here are core interior. Remember the properties of core interior angles. These are the ones that are found inside, which add up to 180. So it means this angle and this angle must add up to 180 degrees.
This angle and this angle must add up to 180 degrees. So these are the properties now that you're putting into consideration of a parallelogram. Because you know that they are parallel, opposite sides. So you see that these are the major properties that you're going to consider, even if you are talking of a rhombus. Talk of a rhombus.
The major part there is of being parallel. Just like a square. It is simply a square.
That has been shifted just a simple square. But now it is no longer in this format, just like this. All sides being equal, but the opposite sides parallel, just like what we saw before.
Opposite sides parallel. This time, they are not just parallel. All sides will be equal in length.
So it is simply a parallelogram with equal sides. Because this here is the same property that we had on a parallelogram. But what we simply have this time is that it is no longer exactly because these two sides were the ones that are equal.
These ones also equal. Like opposite sides, opposite sides, they are equal. Opposite sides are equal.
But on a rhombus, all sides will be equal. All, all, all of them. This one to this one to this one. That is what is now differentiating or what is separating a rhombus from a parallelogram.
So we can even define this or say it is a rhombus. It is, sorry, it is a parallelogram. It is a parallelogram, but with equal sides.
That is the major part. So it is a parallelogram with equal sides. If it has equal sides, it is now a rhombus.
All right. The last one that is also important in our syllabus that has to do also with the properties or that has to do with our lines. It is referred to as a trapezium.
We're going to talk about this as much as we can. A trapezium. A trapezium cannot be related to a parallelogram. It cannot be related to a rhombus. Meaning to say, if it cannot be related to this, it cannot also be related to a parallelogram.
It cannot be related to a rectum. It cannot. be related to a square if it cannot be related to any of these that's a shape on its own this one we call a shape on its own you just need to know its properties you cannot relate it to any shape you cannot say it's a square that has been this you cannot say it's a rhombus that has been no it's a shape on its own other shapes can be related to a parallelogram but this one cannot it's a shape when it's on. what is it that we have it's simply a shape of this nature where we have a one pair opposite sides parallel one pair just one pair just like this one pair only of opposite sides parallel these two opposite to each other they will be parallel that's only only only the major part that you have that's only that so this one you need to know it you need to know it by by heart because it's you cannot relate it from a any shape to say uh when i had a parallelogram i had this but this time this one is on its own only one pair so we consider in this case one pair so it only has uh one pair of opposite sides one pair of opposite uh size are parallel so there will be parallel only one pair only one so it can be given in this manner there are still a trapezium okay you can have it like this okay sorry guys for this i think my network is just being somehow something like this still a trapezium one pair of parallel sides so meaning to say If these two sides, they are parallel, we are now back to the properties of what? Lines, parallel lines.
Meaning to say, this line that we are seeing, it is now the transversal. And this angle and this angle are now all interior angles. We are now back to the properties of parallel lines.
Are you seeing that? These two lines, they are parallel. So what about this line, transversal again?
So what does it mean about these two angles? All interior angles. So that is the purpose of the major purpose of these three shapes.
They have the properties of the parallel line concept. We still have the parallel line concept. The parallel line concept is still there. So let's see.
This is what we have. We are given. In this exercise, which has to include the properties of triangles and quadrilaterals, like I said, number one, calculate the sizes of angle one to angle six from angle one up to angle six.
As we can see, we've got the parallel lines and also we have got a triangle that we are given. All right. That is A, B, C. This was supposed to be C here at this point. we are supposed to have C, like A, B, C, then D, K, and T.
So there are angles that are given there. Angle 1, angle 2, angle 3, angle 4, angle 5, angle 6. All right, let's figure out how can we have these angles. A, B, C, this triangle. If we are given this indication, it is a very, very important indication, these two here.
What are they showing us? They are only on these two sides, side AC. and side b c it means two sides are equal if there was another one like this here it means all the three sides are equal but that is only on these two sides this one what does it represent this one it's for parallel this one it is the one that is going hand in hand with this one to show that this line and this line are parallel but these two that we are seeing um i mean this these ones on the line indicated it can be one like this.
But if they are indicated on a line, it's to show that the lines are equal. But these ones, the arrows like this, is to show that the lines are parallel. So do not confuse with that. So it means we are having the parallel lines.
Okay. Sorry, this was all C. We are having the parallel lines. We are also having two lines which are equal. So this triangle becomes an isosceles triangle where two sides are equal.
Remember this isosceles triangle. So as I say that, if two sides are equal, therefore the base angles, the angles here that are opposite to those sides will be equal. So considering that you are seeing a triangle like this, know that the angles opposite this one and this one opposite to these sides, they are equal. We are talking about what?
This angle 5 and angle 6. They are equal. So by that, we can simply understand that angle 5 is equal to angle 6. Guys, we are just analyzing. We are not answering anything.
Analyze your diagram. All right. That's what we can do.
All right. Let's answer our question from this. All right. How can we find angle 1, K to C?
to t like this this is a straight line and what is it that we understand about angles on a straight line i talked about that angles on a straight line end up to 180 degrees so meaning to say we can calculate angle one from the one that we are given because they are on a straight line so meaning to say uh angle one our angle one was simply going to be 180 minus 32 these two there are two angles on a straight line so simply subtract the other one from what from 180 degrees so it's 180 minus 132 degrees you subtract the 132 degrees, this one, that is also on the line with the angle one. Remember, these two, they are on a straight line. So you subtract another one from 180, the one that you are given.
By doing so, you are obtaining an angle that is the other one on a straight line. Remember, these are simply adjacent supplementary angles. Adjacent supplementary angles. So that is going to be 48 degrees if you subtract. So these two angles, they are adjacent to each other and they are supplementary.
They add up to 180, meaning to say 132 degrees plus angle 1 is equal to 180. So you can simply take 132 to this side to be subtracted from 180. All right. So also try by all means to indicate the angles that you have calculated. I just pay pencil you indicate so that it can help you to determine another one.
So as we can see, we have got angle 1, which is 48 degrees. Okay, what can we do to determine angle 2? Let's go back again to the properties of the lines.
K to T like this, that one, it's a straight line. D to A like this, it's a straight line again. Remember the property of this X.
Whenever you see this X, what do you think about? vertically opposite angles being equal. These two angles being equal. These two angles being equal.
I talked about that. The part of the vertically opposite angles. So it can be of use in that case, the vertically opposite angles.
So meaning to say, using the vertically opposite angles, this angle here, angle one and angle two, they are equal. These are vertically opposite angles. This angle and this angle.
They are equal. They are talking about vertically opposite angles. There is no calculation that you are going to do.
So thus we have got angle. Angle 2. So angle 2 is equal to 48 degrees. Just like that.
Just like that. From which concept? That is from which concept? Alright.
Let's move on to angle three. Can we determine angle three? All right.
According to what we have is still, it's going to be complicated here because we do not have angle four and angle four cannot be related because it cannot be taken from this 132. No. Remember, vertically opposite is like this. You continue with a straight line, straight like this.
That's a straight line. And this one, it's also a straight line. So these ones are vertically opposite. If we were to use The vertical opposite of 132, take note where 132 is. 132 is this one.
It is a vertical opposite angle to these two angles, these ones. Together, they add up to 132, okay? That is angle 3 and angle 4 together combined the wall of angle 3 and 4 is the same as 132 from this. Please be careful there.
So 132 is not equal to angle 3. It is not equal to angle 4. It is equal to the sum of these two, angle 3 and angle 4. So for now, we cannot determine angle 3. We cannot determine angle 4 separately because the wall of this is this 132. All right. Let us just write it as a statement because we can't determine it right now. So we're just going to write as a statement.
Remember what I did? Angle 5 and angle 6, they're equal. So this time we are saying. the wall of this angle angle three plus angle four is supposed to give us the whole angle that we are given there that is the vertically opposite vertically opposite this one to this one so that is a 132 degrees okay so it must give us 132 just leave it like that we can't you You can't give anything.
Yeah, I'm just trying to show you guys that. Because some of us will be following, I have to find angle 1, angle 2, angle 3. Sometimes you see, okay, I can't determine this angle 3 direct. Leave it like that. Move on to another part. Move on to another part.
Do not force that person to say, okay, I have to, I must find that angle three. You waste time, you waste time until you see, okay, there is no solution that I can have. So leave this part alone. You will see what's going to happen later on.
Let's move on to another part that we can find. Since we can't determine angle three and angle four for now, let us leave these two and move on to these other two. Angle six and angle five.
Can we determine any one of these? By determining one of these, we have determined both because I said here angle 5 is equal to what? Angle 5 is equal to angle 6. So if I have one of these, I'm done. All right. We can determine angle 6 from where?
Remember, these two lines, they are parallel. So this is it. Let's move on from K to this point to this point.
and to the point b like this what is it that you're forming that you're familiar to exact alternate angles remember alternate angles this angle and this angle they are equal alternate angles are equal i talked about that so meaning to say angle two and angle six they are equal they form exact alternate angles so this is gonna be 48 degrees. All right. So we can determine angle six. So angle six is equal to 48 degrees.
That is alternate to what? That is an alternate to angle two from the parallel lines. That is the reason that you can put them. Remember, we said these two angles, they are equal from the isosceles triangle.
We said opposite angles to the sides, the ones that are opposite to these sides. They are equal. Angle 5 and angle 6, we said they are equal. So by having angle 6, we also have what? Angle 5. Because angle 5 and angle 6, they are equal.
So also angle 5 is equal to 48 degrees. Look where we are. Now we can determine angle 3. From where?
Remember a triangle now that... angles in a triangle they do add up to 180 degrees we talked about that angles in any any triangle they add up to 180 so by having this now we can take it as an advantage to say angle 3 can be calculated from the angles inside of a triangle since the angles inside of a triangle they add up to 180 it means We can determine our angle, our angle 3. So angle 3 is going to be 180 degrees minus these two angles that are inside the triangle so that we have the remaining one. So you subtract these two angles, the 48 degrees plus the 48 degrees together. Or you just subtract it this way, 180 minus 48. You subtract another 48 again. It is one and the same thing.
The Sim. presentation so this was going to give us what uh angle three all right so if we subtract 180 minus 48 uh minus 48 it was gonna be 84 degrees so that's 84 degrees there thus we have got what angle three so therefore angle three is 84 degrees so here we have got 84 degrees that's what we have day. 84 degrees so finally we can determine uh the angle here uh angle four by using the concept remember i said angle three and angle four they add up to 132 degrees we can use that concept it's up to you because that is from the vertical opposite angles that we have so meaning to say we can simply subtract that uh from what from 132 because angle three and angle four they add up to 132. So to obtain angle 4, it is simply going to be 132 degrees minus 84 degrees.
There are so many ways. Okay, I'm going to explain another one. You can subtract like that, which was going to give us 48 degrees.
Another way to determine the same angle 4 here was to use angles on a straight line here. This is a straight line now. So these two angles, I mean, these three angles are on a straight line.
angle two angle three angle four are on a straight line so they add up to one eight so if i add these two angles all these three angles they must they're supposed to give us 180 degrees. So we can simply subtract to determine what we have angle for. We can simply subtract these other angles that are also on a straight line with angle for. Which is what? Which is the 48 degrees.
We also subtract the 84 degrees. We were going to obtain 48 degrees, which is still the same answer. Or another way was simply to take it this way. These lines.
They are parallel, remember. So we are forming a Z. Look here, parallel line concept.
These lines, we are forming a Z. Even this is just looking another side. That's a Z. Alternate angles. This angle 4 and this angle 5, they are equal from our alternate angles.
So many ways. So at the end, we can conclude that angle 4 is equal to what? 48 degrees. We can use...
the alternate angles concept we can use the angles on a straight line we can use the angles on a straight line this way we can use that part of vertically opposite angles concept to say these two angles they are equal to 132 when added together so you can subtract that from what from 132. so as long you see geometry guys as long it has to do with the geometry It's something that is very, very broad, very, very, it's big. There's a lot that you need to work on. A lot of things that you need to consider, all giving the same answers. So which one do you understand? Work with that.
Make sure you revise that as much as you can. All right. So that was our question.
We calculated all the angles that are given. from angle one up to angle six. Question number two, we are given RSTU is a trapezia.
Calculate the size of angle T and R. It's a trapezia. Remember, I said a trapezium, it's just a parallelogram of its own.
I mean, it's a quadrilateral of its own. A quadrilateral that has nothing to... consider to to any other shape you cannot consider it to a parallelogram it is just on its own a pair of what of parallel sides that is what is important guys if you understand that these two lines are parallel this is your mathematics guys work it the way that you want if you extend this line from u to t like this line i want you to see something that is just clear here if you extend this line like this If we extend this line like this, these are the same thing that we are using. They are parallel.
So what is it that we understand if these two lines are parallel? What do we understand about this line, this one? That's a transversal line. So what do you understand about these two angles, angle R and angle U?
Core interior, they form a C. Remember our C angles, the ones that form a C. Core interior angles.
Add up to an 8. You're talking of the core interior angles there, which add up to 180 degrees. So meaning to say U and R are core interior angles. The same thing happening between T and S. They're also core interior.
Look, they're forming a C like this. That's a C, core interior angle. So all this part, it was all about the core interior angles. So we can determine angle T since core interior angles add up to 180. so angle t is simply going to be 180 degrees minus this other one of 112 degrees just like that the core interior angles concept angles forming a c they are forming a c it's either this way or this way we are forming a c so minutes we can use that concept so 180 minus um 112 was going to give you 68 degrees.
So that's our angle T. Then angle R, you can do the same thing. But this one from the U, right? These are the ones that are within the parallel lines. Remember, from the parallel lines, do not take your C this way.
No, that's not it. You take your C from the parallel lines. These lines, these ones, they are not parallel.
This one, RU and TS are not parallel. The C is like this from the lines that are parallel. These are the ones that create a C. So our C, as we can see, it's already created here. So it's at U and at R.
So you subtract 143 to determine our angle R. So that's from 180 degrees, subtract 143 degrees. Remember, core interior angles, they add up to 180. They are not equal. No, they add up to 180 degrees.
So that was going to give us 37 degrees. Okay, so that is 37 degrees. as our angle as our angle r just like that just like that parallel lines concept played at all all right number three we have got a rhombus so still we'll consider the same properties of our parallel lines guys the opposite sides are parallel and equal but what is important is of the parallel lines there okay these two sides are parallel These two sides are also parallel.
So from that, we can answer our question. Calculate the sizes of angle JML, all right? That is from J to M to L, all right?
The wall of this angle, JML, like that. The wall of this, that is M1 and M2 combined. That is our JML, this one. So which property are we going to use?
Remember, they are parallel lines. These two JM and KL are parallel. So are we seeing that we are forming a C there?
This one, that's a C that we are forming. So as long as we are forming a C, we're talking about what? The core interior angles.
So the angle of 102 and this angle JML must add up to 180 degrees. So that means angle JML is equal to 180 degrees minus 102. Why? They are core interior angles. angles that we are referring to so that was going to give us uh 78 degrees so meaning to say the whole of this angle combined is 78 degrees m1 and m2 together m1 and m2 together it's 78 degrees okay then what about angle m2 m2 what is going to happen is that these two and this line remember this is a rhombus where all the sides are equal so this line what it does it divides these angles into two equal parts it's a special line which divides into two equal parts to divide into two equal parts is called to bisect so we say km bisect this one KM bisects angle JML. It divides this angle into two equal parts.
This KM is a diagonal, a diagonal. Okay. So it divides on a rhombus.
It's going to divide into two equal parts like that. So meaning to say angle M1 and angle M2, they are equal. M1 is equal to M2. So if M1 and M2 are equal and the wall of this angle is 78 the wall of this is 78 i how are you going to determine one of these if they are equal you simply have to divide the 78 degrees by what by two we are talking about two angles that are equal two angles that are equal so you're going to simply divide that was going to be a 39 degrees so meaning to say we have got angle m1 as 39 degrees We have got M2 as 39 degrees. So we have calculated angle M2.
All right, what about angle K1? That is K1, this one. The lines, they are parallel.
This line JK and the line ML, they are parallel. Angle K1 is actually forming alternate angles with angle M2. We are forming a Z.
Look at the Z there. from our parallel lines we are forming a z in that case so meaning to say if this angle is 39 here k1 is going to be also 39 degrees all right we are forming z alternate angles so that is our angle k 1 is equal to 39 degrees that is alternate to what to angle m2 so this is alternate to angle m2 this one angle m2 we are forming as that they are alternate angles So, I always think, guys, we can answer a lot of questions with these geometric reasons, the shapes that we are given. We can answer a lot of questions. Then we are given a question number four.
That is, A, B, C, D is a parallelogram. Same as a rhombus. A parallelogram is just same as rhombus. Only that a rhombus, the sides are equal. But the condition of, because here we're not talking about the sides.
We're talking about. being parallel. So in terms of being parallel, when you see a parallelogram, you have seen a rhombus, whatever that you're answering on a rhombus is the same because these two sides, they are parallel to each other. These two sides, they are parallel to each other. So how are we going to calculate the sizes of this angle?
ADB, ABD, angle C and angle DBC. All right, let's see what is easier for us. Angle ADB.
that is from a to d to b all right is there anything that we can consider from this angle like okay we do not have like anything that you can consider so just try to fill in also it's another way of answering like what is that you can find remember these two lines are parallel let's create a z there is a z alternate angles so angle d this one and this angle b they are equal alternate angles this will be 20 degrees so meaning to say we have calculated angle a b d a to b to d okay just like that try to fill in what you can feel so we've got angle a b d which is 20 degrees alternate to angle this one 20 what is it now that we consider we are inside a triangle there are two angles inside a triangle guys Angles in a triangle, they add up to 180 degrees. So it means we can calculate the remaining angle ADB. That is from A to D to B. Now we can calculate that. Angles inside a triangle, they add up to 180 degrees.
So do subtract these two angles that are also inside the triangle with another one that you need to calculate. So you're going to subtract the 82 degrees. we subtract also The 20 degrees from 180, just like that, from 180, we subtract both angles, the 82 and the 20 degrees that we are given there. That was going to give us 78 degrees, just like that. So we must be able to understand properties of the shapes.
then we used this one we use the properties of we this one this one we use the properties of a triangle angles in a triangle they end up to 180 degrees so meaning to say this angle is going to be uh 78 degrees okay uh let's move on to another one that we are given also from the parallel look here these two lines they are parallel so using this concept here we can go like this this one like this let's go like this let's go produce a z we have a z alternate angles like n you can even remember z or n so this angle at d and the angle at b this one they are equal look where the z is this one from the lines that are parallel these ones so meaning to say if this is 78 degrees this is also going to be uh 78 degrees are we given that uh that is the dbc right we have got d to b to c so we have calculated angle d b c all right so angle d b c was going to be uh 78 degrees alternate to angle a d b they are alternate to each other they give us a z all right let's find the last angle which is angle c inside a triangle big now properties of a triangle so inside a triangle what is it that we consider angles in a triangle that up to 180 or you can take the property of a parallelogram which states about the opposite sides these ones being equal the opposite i mean the opposite angles being equal to say if this is 82 degrees this will be 82 degrees these are properties that you need to consider of a parallelogram but if you do not understand that do not worry because these properties there are so many this is a triangle right D, B, C. It's a triangle. This one. What is it that you know about angles in a triangle?
Angles in a triangle, they add up to 180, just like what we saw before. So we are simply going to subtract the two angles inside of the triangle. This one inside of triangle DBC that we want to remove, which is the 20 degrees and the 78 degrees that we are given. In that case, you are still going to obtain the same 82 degrees, just like I explained before.
So the angle at C was going to be. 82 degrees. So we have calculated or stated all angles that we are given there. All four angles are there. So properties of shapes together with the properties of lines they play a role hand in hand.
So as you revise your lines, revise also the shapes. So do a recap. Just to go through the basics, start by the basics. Understand the properties of each shape.
As you are answering, you are using those properties. But what is important, as we have seen from all these properties that we have stated, it's all on the parallel lines. The major part is about the parallel lines. If you consider the lines that are parallel, that's the major part.
If you consider the shapes that we have, the major part is on being parallel. Here, it's on being parallel. A parallelogram being parallel. That is the major part that we are going to consider.
So, it is only a trapezium which has one pair of parallel sides. This other part is not parallel. But on a parallelogram, all. two pairs consider also on a rhombus two pairs again so these two they work hand in hand the same way that you just worked with a rhombus is the same way that you're going to work with a parallelogram so let's revise as much questions as we can uh like uh considering the parallel lines not any not anything that has got to do with the diagonals no do not consider anything about parallel lines that's what i'm trying to say about these two when it is inside the angles we saw that a rhombus it can bisect the angles into two equal parts which is not happening on a parallelogram this one does not bisect okay but it just gives us that the angles opposite will be equal this angle will be equal to this one. This angle here will be equal to this one.
So opposite angles are equal in a rhombus. Opposite angles are also equal. So guys, there are a lot of things that you can put into consideration to answer these questions. Let us just revise as much as we can.
Then we'll see that we do not have a limit to answer this question. As long as we are talking about the geometry part, there's a lot. So revise the basics.
what you understand use that to answer those questions use the concept that you do understand