Answer
Verified
423.6k+ views
Hint: Join $A$ to $O$. Apply the given condition to the isosceles triangle and use similarity criterion.
According to given data we have 3 conditions,
$AB = AC \to (1)$
$OB$ is the bisector of $\angle B$
So,$\angle ABO = \angle OBC = \dfrac{1}{2}\angle B \to (2)$
$OC$ is the bisector of$\angle C$
So,$\angle ACO = \angle OCB = \dfrac{1}{2}\angle C$$ \to (3)$
Case-1
So, here we have to prove $OB = OC$
Proof:
Now by using condition (1) we can say that,
$AB = AC$
From this condition we say that
$ \Rightarrow \angle ACB = \angle ABC$ [ Where we know that Angles opposite to equal sides are equal]
$\
\Rightarrow \dfrac{1}{2}\angle ACB = \dfrac{1}{2}\angle ABC \\
\\
\ $
$ \Rightarrow \angle OCB = \angle OBC$ [From (2) and (3)]
Hence,
$OB = OC$ [Sides opposite to equal angles are equal]
Hence proved that $OB = OC$.
Case-2
We have to prove that $\angle OAB = \angle OAC$
By using case (1) we know that $OB = OC$
And also from $\Delta ABO$ and$\Delta ACO$, we have
$ \Rightarrow AB = AC$ (Given)
$ \Rightarrow AO = OA$ (Common)
$ \Rightarrow OB = OC$(From (case 1))
$\therefore \Delta ABO \cong \Delta ACO$ (By SSS Congruence rule)
$ \Rightarrow \angle OAB = \angle OAC$ (CPCPT Theorem)
Hence we have proved that$\angle OAB = \angle OAC$.
CASE - 1 CASE - 2
NOTE: In this problem given construction is mandatory to prove the given statements so join $A$ and $O$ points.
According to given data we have 3 conditions,
$AB = AC \to (1)$
$OB$ is the bisector of $\angle B$
So,$\angle ABO = \angle OBC = \dfrac{1}{2}\angle B \to (2)$
$OC$ is the bisector of$\angle C$
So,$\angle ACO = \angle OCB = \dfrac{1}{2}\angle C$$ \to (3)$
Case-1
So, here we have to prove $OB = OC$
Proof:
Now by using condition (1) we can say that,
$AB = AC$
From this condition we say that
$ \Rightarrow \angle ACB = \angle ABC$ [ Where we know that Angles opposite to equal sides are equal]
$\
\Rightarrow \dfrac{1}{2}\angle ACB = \dfrac{1}{2}\angle ABC \\
\\
\ $
$ \Rightarrow \angle OCB = \angle OBC$ [From (2) and (3)]
Hence,
$OB = OC$ [Sides opposite to equal angles are equal]
Hence proved that $OB = OC$.
Case-2
We have to prove that $\angle OAB = \angle OAC$
By using case (1) we know that $OB = OC$
And also from $\Delta ABO$ and$\Delta ACO$, we have
$ \Rightarrow AB = AC$ (Given)
$ \Rightarrow AO = OA$ (Common)
$ \Rightarrow OB = OC$(From (case 1))
$\therefore \Delta ABO \cong \Delta ACO$ (By SSS Congruence rule)
$ \Rightarrow \angle OAB = \angle OAC$ (CPCPT Theorem)
Hence we have proved that$\angle OAB = \angle OAC$.
CASE - 1 CASE - 2
NOTE: In this problem given construction is mandatory to prove the given statements so join $A$ and $O$ points.
Recently Updated Pages
Three beakers labelled as A B and C each containing 25 mL of water were taken A small amount of NaOH anhydrous CuSO4 and NaCl were added to the beakers A B and C respectively It was observed that there was an increase in the temperature of the solutions contained in beakers A and B whereas in case of beaker C the temperature of the solution falls Which one of the following statements isarecorrect i In beakers A and B exothermic process has occurred ii In beakers A and B endothermic process has occurred iii In beaker C exothermic process has occurred iv In beaker C endothermic process has occurred
The branch of science which deals with nature and natural class 10 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
Define absolute refractive index of a medium
Find out what do the algal bloom and redtides sign class 10 biology CBSE
Prove that the function fleft x right xn is continuous class 12 maths CBSE
Trending doubts
Assertion CNG is a better fuel than petrol Reason It class 11 chemistry CBSE
How does pressure exerted by solid and a fluid differ class 8 physics CBSE
Number of valence electrons in Chlorine ion are a 16 class 11 chemistry CBSE
What are agricultural practices? Define
What does CNG stand for and why is it considered to class 10 chemistry CBSE
The rate of evaporation depends on a Surface area b class 9 chemistry CBSE
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
State whether the following statement is true or false class 11 physics CBSE
A night bird owl can see very well in the night but class 12 physics CBSE