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Answers
Answer:
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Step-by-step explanation:
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Step-by-step explanation:
Given :-
In an Isosceles triangle ABC with AB = AC ,the bisectors of ∠B and ∠C Interest each other at O.
To find :-
Show that :1) OB = OC 2) AO bisects angle A
Solution :-
Given that :
∆ ABC is an Isosceles triangle.
AB = AC
=> ∠C = ∠B
=> ∠B = ∠C ------------------(1)
Since The angles opposite to equal sides are equal.
On dividing by 2 both sides then
=> ∠B/2 = ∠C/2
the bisectors of ∠B and ∠C Interest each other at O.
=> ∠OBC = ∠OCB ----------(2)
and
∠B = ∠OBC +∠OBA
and
∠C = ∠OCB + ∠OCA
So,
∠OBA = ∠OCA -----------(3)
OB = OC ---------------------(4)
The sides opposite to equal angles are equal
and
In ∆ AOB and ∆AOC
We have
AB = AC ( Given )
∠OBA = ∠OCA ( from (3))
OB = OC (from (4))
By SAS Property
∆AOB =~ ∆AOC
∆AOB and ∆AOC are congruent triangles
So,
∠BOA = ∠COA (by CPCT)
=> AO bisects ∠A
Hence, Proved.
Used formulae:-
- Opposite angles to the equal sides are equal.
- Opposite sides to the equal angles are equal.
- In two triangles , The sides and the included angle are equal to the sides and included angle of the second triangle then they are congruent triangles and this property is called SAS Property.
- CPCT - Congruent parts in the Congruent triangles.
