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AS Level Mathematics 9709 | Circular Measures 3 | Angle & Triangle Properties | Raja Shafqat — Transcript

by Math with Raja Shafqat · 4,389 words · 615 segments · language en · Watch on YouTube

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0:08Assalam-o-Alaikum, beta. In the last
0:11class, we studied the geometrical
0:12properties of a circle. Today, we are
0:14discussing one more small point.
0:17Obviously, this point is also from
0:18O-Level math. Clear, beta? We call
0:21these angle properties. Basic angle
0:23properties. Now, what are these basic
0:24angle properties? There are three or
0:26four properties that we require. Now
0:28let's check them one by one. The first
0:30property—all these properties are
0:32from O-Level math. Well, whenever we
0:34talk about O-Level math, let's start
0:36with another property. Suppose this is
0:38a straight line. Okay? Suppose this is
0:40a straight line. Okay? And the angle
0:42given to you on this straight line, the
0:45name of this angle is angle A. Come on,
0:47the angle given to you, let's name this
0:50angle as angle A. Clear? If this is
0:53angle A, and I'll call this angle B.
0:56Let's name this angle as angle B. Okay?
0:59We named one angle as angle A and the
1:02other as angle B. Okay? Let me explain
1:04it again. Let me explain it again that
1:07if we have this angle A. If this angle
1:10is A, then this will be angle B. So,
1:12these two angles are called adjacent
1:14angles. Adjacent. So, what did we learn
1:17in O-Level math? Adjacent angles on a
1:19straight line are supplementary.
1:22Meaning their sum is always what? 180.
1:24So we learned in O-Level math that
1:25whatever angles are on a straight line,
1:27their sum is always what? 180. What do
1:30we call this in math language? In math
1:33language, we call it adjacent angles,
1:37adjacent angles, adjacent angles on a
1:41straight line. Adjacent angles on a
1:45straight line. Adjacent angles on a
1:49straight line are supplementary. This
1:52is a very basic condition that a child
1:54learns in very basic math. We call this
1:57condition that adjacent angles on a
1:58straight line are always supplementary.
2:01Supplementary means their sum should
2:03always be 180. If I tell you this angle
2:05is 60, then this angle will obviously
2:07be what? 120. So that the sum of these
2:10two is what? The sum of these two
2:12should be 180. The sum of these two
2:13should be what? 180. So it is a very
2:15basic condition. We call it adjacent
2:17angles on a straight line are
2:18supplementary. Supplementary means
2:20their sum should always be what? 180.
2:23Clear? We used to study one more
2:25property in O-Level math. We used to
2:28call it vertically opposite angles.
2:30Vertically opposite angles. Look, we
2:31used to show this like this too. Then I
2:33will plot its diagram and show you. We
2:36studied in O-Level Math that vertically
2:39opposite angles are equal. Vertically
2:43opposite angles are equal. Okay, now
2:47what was this property? Look, it's a
2:50small property. By the way, this is
2:52Class 9 Math. Suppose you have a
2:54straight line here. Okay? A straight
2:56line. I plotted another line on the
2:58straight line that is intersecting it.
3:00If I tell you, hey, this angle is angle
3:03C. Let me label it for a minute. This
3:06angle is your angle C. And let me label
3:08this angle. This angle is your angle D.
3:11Okay? One I labeled C, one I labeled D.
3:14And if I label this angle as angle B,
3:16okay, there you go, I labeled this
3:18angle. B and I labeled this angle as A.
3:22So what did we learn in O-Level Math?
3:24In O-Level Math, we learned that if
3:26this angle, let's say it's angle C,
3:28then angle C equals what angle? It is
3:30D. We used to say in O-Level Math that
3:33angle C = angle D and we used to say
3:35angle B = angle A. Let's see a simple
3:38proof for this for a minute. If this
3:41angle is 60, then obviously this angle
3:43on the straight line will be 180-60 =
3:44120, which we studied in the previous
3:46question. If this is 60, then this is
3:48120, so then this is also a straight
3:50line. If this is 120, then this will
3:52also be 60. And then this is also a
3:54straight line. If this is 60, then this
3:56will be 120. So technically, these two
3:59angles became? The same. And
4:00technically, these two angles became?
4:02The same. So basically, vertically
4:05opposite angles were derived from
4:07adjacent angles. Derived from adjacent
4:11angles. So these two angles will be
4:13equal, C and D, and these two angles
4:15will be equal, A and B. So we call
4:17these angles vertically opposite angles
4:19. Vertically opposite angles. We used
4:21to call these X-shape angles in O-Level
4:23Math too. Meaning these two angles are
4:25equal and these two angles are equal.
4:27So we used to call these X-shape angles
4:30in O-Level Math too. Used to call them
4:34X-shape angles. So this is also Class 9
4:37Math, basic adjacent angles on a
4:39straight line, meaning those angles
4:41which are on a straight line, both are
4:43equal angles. Both are next to each
4:45other. So their sum is always 180. And
4:47the second property was O-Level math's
4:49vertically opposite angles. Vertically
4:51opposite angles. Condition number three
4:54. Now we will study two things here.
4:57Condition number three was called
4:59alternate angles and corresponding
5:01angles in O-Level math. Alternate and
5:03corresponding. Okay? We are studying
5:05two more properties. In math
5:07terminology, we also call these
5:10properties angles formed by parallel
5:13lines. In math, we also call them
5:17angles formed by parallel lines. Angles
5:22formed by parallel lines. We also
5:24studied this in O-Level math. Obviously
5:26, all these conditions are from O-Level
5:27math. Angles formed by parallel lines.
5:31Look at how. Suppose you have this line
5:34. And this is another line. You see,
5:38these two lines are parallel. I drew
5:40these two lines parallel. Okay? I drew
5:43these two lines parallel. Here, I drew
5:45both lines parallel. Let's copy these
5:48two lines here. Wait a minute. Here.
5:51Let's get some more space. These two
5:53lines are parallel. Now suppose this
5:55line is cutting these two lines. Okay?
5:58Okay? Like this, I randomly drew a line
6:00that is cutting both these lines. It is
6:02cutting both these lines. Now look, if
6:04I tell you this angle is A, then you
6:05would say this angle is also the same.
6:07Let's name it A. Let's name this one C.
6:09So according to the previous condition,
6:12these two angles are equal. According
6:15to the previous condition. Clear?
6:16According to the previous condition. If
6:18I tell you this angle is D, then you
6:20would say this angle is also D. Let's
6:22name it B. Let's name this one D. So
6:24that means these two angles are equal
6:26according to the previous condition.
6:29These two angles are equal according to
6:31the previous condition. Clear, dear?
6:33Okay. So these two were equal by
6:35vertically opposite. These two were
6:36equal by vertically opposite. If this
6:38was 60, we know this is also 60. If
6:40this was 120, we know what this will be
6:42? This will also be 120. Okay, now I am
6:44telling you something new. Now suppose
6:47I am naming this angle E. I named this
6:50angle E. I named this angle F. Let's
6:53just mark it randomly. I named this
6:56angle F. I named this angle G and this
6:59angle H. Here is G and here is H. G and
7:02H. Okay, so now we have studied two
7:05conditions. Now look, the line
7:07intersecting is intersecting these two
7:09parallel lines. That is why, what is
7:11the condition? Angles formed by
7:13parallel lines. So, A and C were equal.
7:16B and D were equal. Similarly, E and F
7:18will be equal. Will G and H also be?
7:19They will be equal. So now look at the
7:22first condition; if this is 60, then
7:24obviously this is also 60, clear kiddo?
7:27Now we used to study a condition in O
7:29Level Maths. Look here, the Z shape
7:31formed here, there it is. So if I say,
7:34what is angle C? If this is 60, then
7:36this will also be 60. So we kept saying
7:39that angle C and E are both equal.
7:41Because what shape was formed here? A Z
7:43shape was formed. So we kept calling
7:45these alternate angles. In O Level
7:48Maths, we kept calling them alternate
7:50angles. Alternate angles, alternate
7:52angles are equal, and in O Level Maths
7:54we also used to call it this. There was
7:57a topic in O Level Maths called '
7:58Bearings'. Bearings, in math they also
8:01called this 'Z shape angles', 'Z shape
8:03angles'. So look, what was formed here?
8:06A Z shape was formed. So if this is C,
8:08then C and E will both be what? The
8:10same. What will C and E both be? The
8:13same. Clear kiddo? I can also read it
8:15like this. This Z shape is also formed
8:17this way, that if this is D, then D and
8:19G will also both be equal. So that
8:21means angle C and E will both be equal.
8:24Angle C = Angle E, so then I can also
8:26say, look, a Z shape is also being
8:29formed here. Here it is, so angle D and
8:31angle G will both be what? The same.
8:33Angle D = Angle G, so in O Level Maths,
8:36we used to call these two angles
8:38alternate angles. We used to call them
8:41Z shape angles. Alternate angles. Look
8:43again. If this is 60, then math says
8:45this is also 60. If this is 60, then
8:47math says this is also 120. This will
8:49also be 120. So if this is 60, then
8:51math says this will also be 60. If this
8:54is 60, then this will also be 60. If
8:56this is 120, then this will also be 120
8:59. Clear, kid? So we used to call these
9:01alternate angles. Alternate, alternate
9:04Z-shape angles. Alternate angles,
9:06meaning look, these two lines are
9:08parallel. This Z-shape angle is formed.
9:10If you look at it this way, these two
9:11lines are also parallel. So, these two
9:13angles will also be what? The same.
9:15Then we used to use another word.
9:17Corresponding angles are equal.
9:18Corresponding angles. We heard another
9:22word in O-level math; that word was
9:24corresponding angles. We used to hear
9:27another term, corresponding angles are
9:28equal. Just like alternate angles are
9:30equal. Corresponding angles. For
9:32corresponding angles, we used a word in
9:35math. That word, that sign was F-shape
9:37angles. F-shape. Note one thing. These
9:40two lines are parallel. Do you see the
9:42arrow sign? It's parallel. We learned
9:44that if this is 60, then this is 120.
9:47Similarly, if this is 60 and 120, then
9:49this was also 60 and 120. So, will
9:52angle A and angle E also be...The same?
9:55Angle, angle, I am repeating it again.
9:57This is an F-shape angle. So angle A
9:58and angle E will be equal. And angle D
10:00and angle H will also be equal. So I
10:02can say that angle A is equal to angle
10:05E. And I can also say that angle D is
10:08equal to angle H. So we used to call
10:11these angles corresponding angles.
10:14Corresponding, so we kept saying that
10:16corresponding angles are equal.
10:18Corresponding angles are equal. In math
10:20, we also called them F-shape angles.
10:22In math, we also used to call them
10:24F-shape angles. Look again. If this is
10:2660, then this will also be 60. If this
10:28is 120, then obviously this will also
10:30be 120. So this same situation will
10:32transfer below too. This is 60. If this
10:35was 60, then this is also 60. If this
10:38is 60, then this is also 60. If this is
10:40120, then this is also 120. Clear, kid?
10:43This is also 120. So we used to call
10:45these angles corresponding angles and
10:46alternate angles. Where are
10:48corresponding and alternate angles
10:49formed? Between parallel lines. Between
10:52parallel lines. Then we also used to
10:54study one more thing. Interior angles
10:56are supplementary. Then we used to
10:58study another condition. What was the
10:59name of that condition? The name of
11:02that condition was interior angles are
11:04supplementary. Interior angles are
11:08supplementary. Supplementary. Well,
11:12look, now let's see the proof for this
11:13too. Supplementary means their sum will
11:16be what? It will be 180. In math, this
11:19was also called C-shape angles. Or in
11:22math, this was also called U-shaped
11:25angles. C-shaped angles or U-shaped
11:27angles. C-shape or U-shape. We keep
11:30calling it C-shape or U-shape. Look how
11:33? Look, these two lines are parallel.
11:35Two minutes ago, I gave an example that
11:37if this is 60, then this will also be
11:3960. If this is 120, then obviously this
11:41will also be 120. This will also be 60.
11:44This will also be 60. This will also be
11:47120. This will also be 120. So you see
11:49that this 120 and this 60, what is
11:51their sum? This is a C-shaped angle.
11:53Look, this is a C-shaped angle. Here it
11:56is. So that means what will be the sum
11:58of E and D? 180. So we used to say in
12:01O-Level math that angle E and angle D,
12:04angle D and angle E, what will be their
12:07sum? It will be 180. So what will we
12:09call these two angles? How are these
12:10two angles? Supplementary. Meaning this
12:12is 120. This is 60. Look, what is the
12:14sum of both? Similarly, I can also say
12:16what is the sum of C and G? The sum of
12:19C and G is also 180. So I can also say
12:22that angle C and angle G, what is the
12:25sum of both? 180. What is the sum of
12:28both of these too? 180. Clear, dear? So
12:31we learned today that this angle and
12:32this angle are both equal. A and E,
12:34both these angles are equal. A and E,
12:36so can we say why they are equal?
12:38Corresponding. Angle D and angle H are
12:40also equal. Why can we say that?
12:42Corresponding. Then look, we learned
12:44about interior angles, the sum of D and
12:47E is 180. Then we studied again that
12:49these two lines are parallel. The sum
12:51of C and G is 180. So we studied four
12:53conditions. Number one, what is the
12:55angle on a straight line always? 180.
12:58Angle on a straight line, meaning
13:00adjacent angles are supplementary. Then
13:02we studied vertically opposite angles.
13:03C and D are equal. B and A are equal.
13:06Vertically opposite angles. Then we
13:08studied alternate angles, Z-shaped
13:10angles. C and E are both equal. Then we
13:13learned that these are also alternate
13:14angles. D and G are both equal. This
13:17became an alternate angle. Then we
13:19studied corresponding angles. Angle A
13:20and E are both equal. Then we learned
13:23that angle D and H are both equal.
13:25Angle B and F are both equal. Angle C
13:28and, sorry, angle B and G are both
13:31equal. Angle C and angle F are both
13:33equal. Clear? So what kind of angles
13:35are these? Corresponding angles. Then
13:37we studied, what is the sum of D and E?
13:39180. Because what shape is being formed
13:41here? C shape. C shape or U shape. Then
13:44we studied what the sum of C and G
13:46would be? This will also be 180.
13:47Because what shape is being formed here
13:49as well? C shape. Okay? C shape or U
13:53shape. Let me explain again. This is
13:55called a C shape or U shape. Here you
13:57go, C shape or U shape. C or U shape.
14:01We studied all these conditions. In O
14:03Level Math. In O Level Math. I am
14:05telling you one more small property.
14:07Then we will start further. So all
14:09these are adjacent angles, vertically
14:11opposite angles, alternate angles,
14:13corresponding angles, interior angles.
14:16So basically, we studied five points
14:18here. Basically, we studied five points
14:21here. Look, in the same way. Now there
14:24are some properties of triangles as
14:25well. Angle properties of triangles. I
14:28am telling you a small sixth point.
14:30Let's read this. Angle properties of
14:32triangles. Take a look at this once.
14:34Then God willing, we will proceed
14:36further. Its name is angle properties
14:39of triangles. Angle properties of
14:42triangles. Okay? This is a small
14:45condition. Angle properties of
14:47triangles. Now let's see what needs to
14:48be done in this. First of all, if I
14:50tell you that, hey, this is a triangle.
14:52I just randomly sketched a triangle.
14:54Here you go, I sketched a triangle.
14:56This is also obviously all O Level Math
14:58. I sketched a triangle. Math says if
15:01you see this is your angle q. Note it.
15:04Name this angle anything. Suppose this
15:07angle is Q. Suppose this angle is A.
15:09Suppose this angle is B and this angle
15:11is C. Come on, here you go. Let's label
15:13it like this, suppose this is A.
15:15Suppose this is B. Suppose this is C
15:16and suppose this is...Q. Let's name it
15:18D instead of Q. Name it anything. Here
15:20you go, this is D. So what did we study
15:22in O Level Math? We studied a property
15:25in O Level Math that the exterior angle
15:27of a triangle. We studied in O Level
15:30Math that the exterior angle of a
15:32triangle is equal to the sum of
15:34interior opposite angles. Well, if any
15:37student has a doubt, I will prove it to
15:39you. Look how, if I say this is 60. If
15:42I say this is 60. You will say this
15:44angle is 120 because of angles on a
15:45straight line. This is 120. Then answer
15:48me one thing. So, if I look at this
15:51triangle, what should the sum of a + b
15:54+ 120 be? 180. So, when that subtracts
15:58over there, what do I find the answer
16:00for a + b is? 60. That means if this is
16:0360, then look at these two interior
16:06angles, which angle is this? Exterior,
16:09and this is its interior angle. The
16:11angle next to the exterior is called
16:12the interior angle. These two are
16:14adjacent angles. So, math says that
16:17this exterior angle is equal to the
16:19opposite interior angles. So, what
16:22property did we learn in O Level Math?
16:25We learned the property in O Level Math
16:28that the exterior angle of a triangle
16:31...we learned in O Level Math. The
16:36exterior angle of a triangle is equal
16:41to the sum of the interior opposite
16:45angles. The exterior angle of a
16:48triangle is equal to the sum of the
16:51interior opposite angles. Listen again.
16:54This is the exterior angle of the
16:56triangle. The outer angle. For a minute
16:58, let me assume. This is 60. If this is
17:0160, this is exterior. Then who is its
17:03interior? 120. Now we know that the
17:06angle sum in a triangle is always 180.
17:10So, if this 120 subtracts over there,
17:13the answer for a + b is 60. Meaning a +
17:16b will indirectly be equal to d. So,
17:20math says that any triangle's exterior
17:22angle is equal to its opposite interior
17:25angles. This is exterior, this is
17:29interior, and the two angles other than
17:32the interior are called interior
17:35opposite angles. Interior opposite
17:38angles. So, there is a property of a
17:40triangle that the exterior angle of a
17:41triangle is equal to the interior
17:43opposite angles. And one thing everyone
17:46already knows is when we talk about any
17:48triangle, what is the interior sum of
17:50the triangle? What is the sum of all
17:52the angles in a triangle? It is 180. We
17:54already know that in a quadrilateral
17:56the sum is 360 and in a triangle the
17:58sum is 180. We already know this. We
18:01already know this. So, what do we call
18:03this? These are the angle properties of
18:05a triangle. Okay, I am telling you one
18:07more small property. Okay? We also
18:10studied this previously in circle
18:11properties. So, I will repeat it once
18:14more so that you remember it. Whenever
18:16we talk about an isosceles triangle in
18:18math, look, here it is. If I tell you
18:21the triangle is isosceles. If these two
18:23sides are equal, then we know that its
18:25two base angles will also be equal.
18:27Clear, son? So we used to say that if,
18:30let's say, I name this triangle ABC,
18:33then what did we say? In O-Level maths,
18:36we used to say that angle B and angle C
18:38will both be the same. We have already
18:41studied this property. I explained the
18:43name of this triangle in the circle
18:45properties. So, what is the name of
18:47this property? This is an isosceles
18:49triangle. What is the name of this
18:52triangle? Isosceles triangle. We have
18:55already studied this thing. The name of
18:57the topic was circle properties. I
18:59explained there too that when a
19:01triangle is isosceles, its two sides
19:03are equal. So its base angles will be
19:05equal. So the most important property
19:07of an isosceles triangle is that when
19:09you drop a line in it, what does it do
19:12to the base? It bisects the base of the
19:14isosceles triangle. Clear, son? When
19:18you drop a line in an isosceles
19:19triangle, what does it always do to the
19:21base? It bisects it. This is a property
19:24of an isosceles triangle. Okay? Of an
19:27isosceles triangle. So I can say that
19:29in an isosceles triangle, AB and AC are
19:31both equal. If these two sides are
19:33equal, then its base angles will also
19:36be equal. If AB and AC are both equal,
19:38then maths says that angle B and angle
19:40C will also both be equal. Angle B and
19:43angle C will both be equal. We studied
19:45this property of an isosceles triangle.
19:47So when you drop a line in an isosceles
19:49triangle, what does it always do to the
19:51base? It bisects it. Bisects. And it
19:54also divides both these angles equally.
19:57I have already explained this property
19:59in circles. Meaning, if I tell you that
20:01this angle is alpha, you will say that
20:02this is also alpha. This is a property
20:04of an isosceles triangle. Meaning, this
20:06is its line of symmetry. This can fold
20:08onto this, and this can fold onto this.
20:11Okay, son? Look at another triangle.
20:13Equilateral triangle. We also studied
20:15this in O-Level maths. Look at this one
20:16too, for once. Because why are we
20:17studying all these things? Because
20:19these are all required. They will be
20:20used in this. So the name of this
20:23triangle is an equilateral triangle.
20:27Equilateral triangle. Because we have
20:30already studied the right-angled
20:31triangle. The name of this point is an
20:33equilateral triangle. Well, we also
20:35studied in O-Level maths that any
20:36triangle whose three sides are equal is
20:38called an equilateral triangle in maths
20:40. Okay? What do we call it in maths?
20:42Let's draw a triangle. Suppose this is
20:45a triangle and all three sides of this
20:47triangle are equal. Suppose we name
20:49this triangle, let's name it. I have
20:53named the triangle ABC, here it is. I
20:55have named this triangle A, B, C. We
20:57have named this triangle A, B and C. So
21:00if the triangle is equilateral, then
21:02equilateral means what will its three
21:05sides be? They will be equal. So what
21:07did we learn in O-Level Math that when
21:09we talk about equilateral, an
21:10equilateral is also called a regular
21:12polygon. So that means if this is angle
21:14B, this is angle C, and this is angle A
21:15, then we know these three angles will
21:17be equal. So what did we learn in
21:19O-Level Math that if a triangle is
21:21equilateral, it means its three sides
21:23are equal. Meaning all three of its
21:25sides are equal. So then we learned in
21:27O-Level Math that its three angles will
21:29also be equal. Angle A, angle B, angle
21:32C, and we know we can say that all
21:35three angles will be the same, and what
21:37will all three angles be? 60. Because
21:41what we did was divide the angle sum
21:43equally. So in two or three questions
21:45in past papers, you will also find
21:47equilateral triangles. You will also
21:49find isosceles triangles and also this
21:51property that the exterior angle of a
21:53triangle is equal to the sum of
21:55interior opposite angles. And we all
21:58know this fact that in any triangle,
21:59what is the sum of the angles always?
22:02The sum of the angles is always 180. We
22:05all already know this. Clear, kid? So
22:07we haven't learned anything new yet. So
22:09this point we studied today is called
22:11geometrical properties, or let's call
22:13them angle properties. Angle properties
22:16. And we studied two things in this
22:19today. Adjacent, vertically opposite,
22:22alternate, corresponding, and interior.
22:26We studied these five properties. Then
22:28I told you about triangles. The
22:30triangle property that the exterior
22:31angle of a triangle is equal to the sum
22:33of the interior opposite angles. I told
22:35you about the isosceles property. Sides
22:37equal, base angles equal. When we drop
22:39a line, what will it do to the base?
22:41Bisect. We can also say that an
22:43equilateral triangle is a pair of three
22:45equal sides. I can say it that way too.
22:48These two sides are also equal to each
22:50other. These two sides are also equal.
22:52So technically, three pairs have been
22:54formed. This and this are equal. One,
22:56this and this equal, two, this and this
22:58equal, three, so technically
23:00equilateral means all sides are equal.
23:03So what happens in an equilateral
23:04triangle? There are three sides. There
23:07are three pairs of two equal sides.
23:09Three pairs of equal sides. Clear, dear
23:11? Alright, let's start the next point.
23:14This is mensuration. The mensuration
23:17related to circles that we studied in O
23:19Level Maths. That mensuration is also
23:21required for this topic. There is one
23:24more point after this. Then finally we
23:26will start our chapter. Mensuration.
23:28Let's start from absolute zero.
23:30Whatever mensuration related to circles
23:33we studied in O Level Maths, we need
23:35all of it. Number one, first of all,
23:38let's start with the circle in the
23:39circle family. We are starting the
23:42circle family with the circle itself.
23:45Now we need a bit of mensuration. What
23:47did we study in O Level Maths? We
23:49studied in O Level Maths, obviously,
23:51that this is a circle. Clear? We
23:54studied in O Level Maths that this is a
23:56circle. Alright? In the next class,
23:58InshaAllah, we will continue from here,
24:00dear. Okay?

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