Question

 


                                

24. O is a point in the interior of ∆ ABC , such that BO and CO are the bisectors of ∠B and ∠ C respectively . Find ∠ A if ∠ BOC = 1350 .





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Answer 1

$\begin{gathered}\begin{gathered}\bf \:Given - \begin{cases} &\sf{O \: is \: point \: in \: interior \: of \: \triangle \: ABC} \\ &\sf{BO \: is \: angle \: bisector \: of \: \angle \: B \: }\\ &\sf{CO \: is \: angle \: bisector \: of \: \angle \: C } \\ &\sf{\angle \: BOC = 135 \degree}\end{cases}\end{gathered}\end{gathered}$

$\begin{gathered}\begin{gathered}\bf \: To \: find - \begin{cases} &\sf{\angle \: A }  \end{cases}\end{gathered}\end{gathered}$

$\large\underline\purple{\bold{Solution :- }}$

$\rm :\implies\:In \: \triangle \: ABC$

• BO is angle bisector of ∠B

$\implies \: \boxed{ \red{ \tt \: \angle \: ABO=\angle \: OBC \: = \: x \: (say)}}$

• CO is angle bisector of ∠C.

$\implies \: \boxed{ \red{ \tt \: \angle \: OCA = \angle \: OCB\: = \: y \: (say)}}$

Now,

$\implies \: \boxed{ \red{ \tt \: In \: \triangle \: BOC}}$

We know,

• Sum of the angles of a triangle is supplementary.

$\rm :\implies\:\angle \: BOC \: +\angle OCB \: + \angle \: OBC \: = 180 \degree$

$\rm :\implies\:135 + x + y = 180$

$\rm :\implies\: \boxed{ \pink{ \bf \:x + y  \:  =  \tt \: 45 \degree}} - - (1)$

Now,

$\implies \: \boxed{ \red{ \tt \: In \: \triangle \: ABC}}$

$\rm :\implies\:\angle \: ABC \: + \angle \: BCA \: \angle \: BAC \: = 180 \degree$

$\rm :\implies\:2x + 2y + \angle \: A \: = 180$

$\rm :\implies\:2(x + y) + \angle \: A = 180$

$\rm :\implies\:2(45 \degree) + A = 180 \degree \: \: \{ \green{using \: (1)} \}$

$\rm :\implies\:90 \degree \: + \: \angle \: A = 180 \degree$

$\rm :\implies\: \boxed{ \pink{ \bf \:\angle \: A   \:  =  \tt \: 90 \degree}}$

Additional Information

Properties of a triangle

• A triangle has three sides, three angles, and three vertices.

• The sum of all internal angles of a triangle is always equal to 180°. This is called the angle sum property of a triangle.

• The sum of the length of any two sides of a triangle is greater than the length of the third side.

• The side opposite to the largest angle of a triangle is the largest side.

• Any exterior angle of the triangle is equal to the sum of its interior opposite angles. This is called the exterior angle property of a triangle.

Based on the angle measurement, there are three types of triangles:

• Acute Angled Triangle : A triangle that has all three angles less than 90° is an acute angle triangle.

• Right-Angled Triangle : A triangle that has one angle that measures exactly 90° is a right-angle triangle.

• Obtuse Angled Triangle : triangle that has one angle that measures more than 90° is an obtuse angle triangle.