Let's do this question who is the center of Triangle Park, you are all right.
Find angle Q or if angle Q is angle or is equal to sixty five degrees.
All right.
Pause the video, give it a try and then let's do it together.
All right.
We're back.
I hope you've given it a try.
Let's do it together.
It's given that these two angles are sixty five degrees.
Right.
Therefore as for the angle some property of a triangle we can see that angle is equal to 180 minus sixty
five plus sixty five.
That's one thirty over here.
And you get angle B is equal to fifty degrees.
All right.
Now over here, always the center and this circle is the inner circle of Triangle Park.
You are therefore these lines you, you are and B are tangents to this circle at these points.
Let's join these points to.
All right.
Now, because P, Q, R and B are tangents to this circle.
Therefore these angles are ninety degrees.
Remember a tangent to a circle.
Meets the circle only at one point, and if you join that point, for example, in this case, this
point to the center right, the angle formed with the center, the point of intersection and the tangent,
this angle over here will be 90 degrees.
All right.
Now, let me mark these three points as a point A, point B and point C.
Now let's check quadrilateral A, B, B, or you can see that learning A, B, B is a kite.
What is a kite?
A kite is a special type of quadrilateral where two sets of Adderson sides are equal or hear it or and
or B are radiuses of the inner circle.
So these two sides are equal and Eppy and B, we are also equal because these are the engines from an
external point to a circle.
Right now, these two things to learn more about it, check the section where we discuss Sakalys in
great depth.
Now, inner circle, if you draw Tandan students are from an external point, like in this case from
point B, which is an external point to this circle.
We are drawing tangents P, Q and B are right.
In that case, we will be equal to be B.
This is a property that we learn in circles now because of that, because these two sides are equal
and these two sides are equal.
Therefore, APB is a kite.
All right.
Let's just note that over here now you can see that this angle is ninety degrees.
This angle is ninety degree.
Right.
And this angle over here, that's angle P is 50 degrees.
Now, we have seen that the sum of the angles of a triangle is 180 degrees.
Right.
So what will be the sum of the angles of a quadrilateral?
If you have any quadrilateral, you can see that you can split it and make it into two triangles for
this angle plus this angle.
Plus this angle gives you a 180 degree.
Similarly, this angle plus this angle, plus this angle gives you 180 degrees.
Therefore, the sum of the internal angles of a quadrilateral is 180 plus 180.
That's 360 degree.
Now you will learn more about this under the section of quadrilaterals.
All right.
I'm just listing it out here because I wanted this to be linked with incentive, which is a topic that
we have learned in this section.
All right, now, because angle B plus angle over here and angle B over here and angle it or B is equal
to 360 degree.
Therefore, I can say that angle all is equal to 360, minus 90, minus 90, minus 50.
Right.
Let me quickly write that over here.
What is it that we've just discussed?
360 is equal to angle.
It will be.
Plus Angle or AP.
Plus Angle or Beebee.
Plus angle, Pete, right?
So these are the four internal angles of this quadrilateral.
Now we know that angle is.
B is what we are trying to find, but that's what we have put us angle or here angle or a B is 90 degrees
angle or B, B is 90 degrees and angle is equal to 50 degrees.
So that's why Angle is equal to 360, minus 90, minus 90, minus 50.
All right.
Let me make some space over here.
Now or here, we get that angle or is equal to 130 degrees, all right, now you can see that quadrilaterals
E or C, Q and B or C, R, that's E or CQ, this one over here, and B or C, R, this one over here.
These are also kite's.
Why is that so?
It's the same reason over here, for example, in AC.
Q Q is equal to QC because these are tangents to a circle from an external point.
Q And is equal to or C because these are in radiuses and the radius is the same.
All right.
Now these are kite's.
Now, let me join you and or OK, now indicate the longer diagonal is an angle by sector.
So that's why I can say that this angle and this angle are equal.
Similarly, this angle on this angular equal now learn more about heights in this section under quadrilaterals.
I'm just listing out here because I don't want it to be completely separated.
In your mind, this is related to topics in quadrilaterals and circles and triangles, right?
We have an instructor over here.
That's something that we've learned in triangles.
We have a concept of Kate over here.
That's something that we learn in quadrilaterals and we have tandem's over here, which we learn under
circles.
So it's a multi concept question.
All right.
Now let's press on.
So AOK, you and are kites and indicate the longer diagonal is an angle by sector.
Let me just quickly explain that over here.
So I have a Kate over here.
This is the shape of a guide.
These two sides are equal.
Now to diagonals can be formed.
This is the longer diagonal, right?
This is the longer diagonal.
This is the shorter diagonal.
Now, the longer diagonal is an angle bisecting.
So this angle will be equal to this angle and this angle will be equal to this angle.
That's the property that we have used over here.
QOL is the longer diagonal of AoE CQ.
That's why Angle A OCU and Angle C or Q are equal and I've marketed as X and X.
Similarly, Anglesey or R and Angle R or B are equal because or is the bigger diagonal of quadrilateral
B or C R.
That's why I've marked as white and white.
Now you can see that X plus X plus Y plus white.
That is two X plus two Y is equal to 360 minus this angle over here, which is angle it will be or we
have marked angle B over here and this angle is one thirty degree.
Right.
Therefore X plus X plus Y plus Y that's two plus two.
Y is equal to 360 minus one thirty which is equal to thirty now therefore we can find that angle Q or
R which is angle X plus angle Y over here.
That would be half of to thirty.
Right.
Or X plus why is equal to two thirty by two.
That gives you your answer as one hundred and fifteen degrees.