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In the below figure, X=62, XYZ=54. If YO and ZO are the bisectors of angle Y and angle Z then find OZY&YOZ.
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Answer
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Hint: We know that the sum of all the angles of a triangle are 180 so using this property we can find the value of the third angle Z of the triangle XYZ because we know the values of angle X and angle Y. After that it is given that OY and OZ are the angle bisectors of angles Y and Z respectively so OYZ&OZY are half of angles Y and Z using this information we can find the value of OZY and YOZ is calculated by using the property of the triangle that sum of the all the angles of a triangle are 180 and here the triangle is OYZ.

Complete step by step answer:
We have given a figure below, in which X=62, XYZ=54.
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We are asked to find OZY&YOZ.
First of all we are going to find the value of angle Z which we do by using the property of the triangle of XYZ that the sum of all angles of the triangle are 180.
X+Y+Z=180
Substituting X=62 and Y=54 in the above equation we get,
62+54+Z=180.......Eq.(1)116+Z=180Z=180116Z=64
Hence, we got the value of angle Z as 64.
It is also given that OY and OZ are the angle bisectors of angles Y and Z respectively. This means that the values of OYZ&OZY are equal to half of angle Y and half of angle Z respectively.
OYZ=12YOYZ=12(54)OYZ=27
From the above, we got the value of OYZ=27.
OZY=12ZOZY=12(64)OZY=32
From the above, we got the value of OZY=32.
Now, to find the value of YOZ we are going to use the property of triangle which says that the sum of the angles of a triangle are 180 in ΔYOZ.
YOZ+OZY+OYZ=180YOZ+32+27=180YOZ+59=180YOZ=18059YOZ=121
From the above, we got the value of YOZ=121.

Hence, we got the value of YOZ=121 and OZY=32.

Note: This question demands the knowledge of the property of a triangle which says that sum of all angles of the triangle are 180 if you don’t know this property then you are lost in this problem and you cannot solve anything in it.
The mistake that could happen in this problem is the calculation mistakes so be careful. Another thing through which you can check whether the angles that you calculated are correct or not by using these angles in the equation of sum of all the angles of the triangle as 180.
For e.g. to check whether OZY=32, first of all double it so that we will get the angle Z.
2OZY=64=Z
Now, substituting the above value of angle Z in eq. (1) we get,
62+54+64=180180=180
As you can see that L.H.S is equal to R.H.S so the value of OZY=32 which we have calculated is correct.
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