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Answer:
Option b
Step-by-step explanation:
Given :-
In ∆ ABC , < A = 50° and the external bisectors of < B and < C meet at O.
To find :-
Find the measure of < BOC ?
Solution :-
Method -1:-
In ∆ ABC ,the bisectors of <B and <C meet at O then <BOC = 90°-(<A)/2
=> < BOC = 90°-(50°/2)
=> <BOC = 90°-25°
=> <BOC = 65°
Method-2:-
Given that
In ∆ ABC ,
< A = 50°
the external bisectors of < B and < C meet at O.
=> < ABC = <CBO
Let < ABC = <CBO = x° ----------(1)
=> <CBD= 2x°
and
We know that
The exterior angle is equal to the sum of the two opposite interior angles
=> <CBD = <A+<C
=> 2x° = 50°+ <C
=> < C = 2x° -50° ------(2)
and
=> <ACB = < BCO
Let <ACB = < BCO = y° ------(3)
=> <BCE = 2y°
=> We know that
The exterior angle is equal to the sum of the two opposite interior angles
=> <BCE = <A+<B
=> 2y° = 50°+<B
=> <B = 2y°-50° ---------(4)
We know that
The sum of the three angles in a triangle is 180°
In ∆ ABC,
< A+<B+<C = 180°
=> 50°+ 2y°- 50°+2x°-50° = 180°
=> 2x°+2y°-50° = 180°
=> 2x° + 2y° = 180°+50°
=> 2x°+2y° = 230°
=> 2(x°+y°) = 230°
=> x°+y° = 230°/2
=> x°+y° = 115° -------(5)
We know that
The sum of all angles in a triangle is 180°
In ∆BOC,
<BOC + <OCB + <CBO = 180°
=><BOC + y°+x° = 180°
=> <BOC +115° = 180°
=> <BOC = 180°-115°
=> <BOC = 65°
Answer:-
The measure of <BOC for the given problem is 65°
Used formulae:-
→ The exterior angle is equal to the sum of the two opposite interior angles in a triangle.
→ The sum of all angles in a triangle is 180°
Step-by-step explanation:
Given:
ABC is a triangle and external bisector of ZB and ZC meet at O, And <A=50°
Then Ex ZPBC =<A + <C
(1)
Then Ex ZQCB =<B + <A ZPBC = 220BC (BO is bisector of angle
B)
ZQCB = 22BCO (CO is bisector of angle C)
Add (1) anbd (2) we get:
<PBC + <QCB = <A + C + 2B + <A → 24OBC+22BCO = <A + 180°
ZPBC + ZQCB = <A + <C + <B+ <A
22OBC+22BCO = <A + 180°
In triangle ABC
ZA + 2B + C = 180°
and
ZA = 50° (given)
Then,
2220BC + 22BCO = <A+180°
→ ZOBC+ ZBCO = 1/2
=> <+90° = 50/2+90 = 115°
ABOC
ZBOC+ZOBC+<BCO = 180°
→<BOC=180-115-65°