5.In an isosceles triangle ABC with AB=AC, the bisectors of B and C intersect each
other at O. show that AO bisects A
Answers
Answer:
In ∆ABC,
➝ AB = AC
➝ ∠B = ∠C .......[Equation (I)]
In ∆BDC,
➝ AB = AC
➝ ∠B = ∠C ......[From Equation (I)]
➝ ½ ∠B = ½ ∠C
➝ ∠OBC = ∠OCB .......[Equation (II)]
➝ BO = CO ......[Sides of an isosceles ∆]
- Considering ∆ABO and ∆ACO :
➝ AB = AC ......[Given]
➝ AO = AO ......[Common]
➝ BO = OC ......[Equation (III)]
⛬ ∆ABO ≅ ∆ACO
➝ ∠BAD = ∠CAD
⛬ ∠A bisected by OA
Answer: Use the result that angles opposite to equal sides of a triangle are equal and its Converse to show part (i ) & show ΔAOB ≅ ΔAOC by using SAS Congruent rule & then use CPCT for part (ii).
____________________________________
____________________________________
[fig. is in the attachment]
Given:
ΔABC is an isosceles∆ with AB = AC, OB & OC are the bisectors of ∠B and ∠C intersect each other at O.
i.e, ∠OBA= ∠OBC
& ∠OCA= ∠OCB
To Prove:
i) OB=OC
ii) AO bisects ∠A.
Proof:
(i) In ∆ABC is an isosceles with AB = AC,
∴ ∠B = ∠C
[Since , angles opposite to equal sides are equal]
⇒ 1/2∠B = 1/2∠C
[Divide both sides by 2]
⇒ ∠OBC = ∠OCB
& ∠OBA= ∠OCA.......(1)
[Angle bisectors]
⇒ OB = OC .......(2)
[Side opposite to the equal angles are equal]
(ii) In ΔAOB & ΔAOC,
AB = AC (Given)
∠OBA= ∠OCA (from eq1)
OB = OC. (from eq 2)
Therefore, ΔAOB ≅ ΔAOC
( by SAS congruence rule)
Then,
∠BAO = ∠CAO
(by CPCT)
So, AO is the bisector of ∠BAC.
Step-by-step explanation: hope it will help you
please mark ema s brainlist