Bisectors of the angles B and C of an isosceles triangle ABC with AB = AC intersect each other at O. Show that external angle adjacent to ∠ABC is equal to ∠BOC.
Solution:
Given, ABC is an isosceles triangle with AB = AC
Bisectors of the angles B and C intersect each other at O.
We have to show that external angle adjacent to ∠ABC is equal to ∠BOC
Line segment CB is extended to a point D outside the triangle.
Considering triangle ABC,
Given, AB = AC
We know that the angles opposite to the equal sides are equal.
∠ACB = ∠ABC
Since OB is the bisector of angle B
∠ABO = ∠OBC
∠ABC = ∠ABO + ∠OBC
∠ABC = ∠OBC + ∠OBC
∠ABC = 2∠OBC ------------ (1)
Since OC is the bisector of angle C
∠ACO = ∠OCB
∠ACB = ∠ACO + ∠OCB
∠ACB = ∠OCB + ∠OCB
∠ACB = 2∠OCB ------------- (2)
So, 2∠OCB = 2∠OBC
∠OCB = ∠OBC --------------- (3)
Considering triangle BOC,
∠OBC + ∠OCB + ∠BOC = 180°
From (3),
∠OBC + ∠OBC + ∠BOC = 180°
2∠OBC + ∠BOC = 180°
From (2),
∠ABC + ∠BOC = 180°
We know that the linear pair of angles is equal to 180 degrees.
∠ABD + ∠ABC = 180°
∠ABC = 180° - ∠ABD
Now, 180° - ∠ABD + ∠BOC = 180°
180° - 180° + ∠BOC = ∠ABD
∠BOC = ∠ABD
Therefore, the external angle adjacent to ∠ABC is equal to ∠BOC.
✦ Try This: In a ΔABC, AD is the bisector of ∠BAC. If AB = 8 cm, BD = 6 cm and DC = 3 cm. Find AC.
☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 7
NCERT Exemplar Class 9 Maths Exercise 7.3 Problem 10
Bisectors of the angles B and C of an isosceles triangle ABC with AB = AC intersect each other at O. Show that external angle adjacent to ∠ABC is equal to ∠BOC
Summary:
Bisectors of the angles B and C of an isosceles triangle ABC with AB = AC intersect each other at O. It is shown that external angle adjacent to ∠ABC is equal to ∠BOC
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