In the following figure, OA = OC and AB = BC.
Prove that:
(i) ∠AOB = 90°
(ii) ∆AOD ≅ ∆COD
(iii) AD = CD

Proof,
In triangle ABO and CBO
AB=CB
BO=BO{commom among triangle ABO , CBO
AO=CO
So, triangle ABO congurent to CBO by SSS
​angle ABO=angle CBO {CPCT}
ANGLE AOB=COB{CPCT}
LET AOB=COB=X
X+X=180
2X=180
X=90
AOB=
90 PROOVED

In triangle ABD, CBD
​AB=CB
ABO=CBO
BD{COMMON}
So triangle ABD congurent to CBD by SAS
AD=CD{CPCT}
​PROOVED






 
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Here AOB + COB is equal to 180 bcz it forms a linear pair

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(1)= AO=BO AB=AO.and AB is the hypotenuse so angle AOB = 90
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In triangle ABO?and triangle CBO

AB=CB

BO is common side

AO=CO

Therefore, triangle ABO?is congurent to triangle CBO?by?SSS rule of congruency.

So, ?angle ABO=angle CBO?(C.P.C.T.)

And, angle AOB = COB (C.P.C.T.)

Let angle AOB = angle COB = x.

Then, x + x = 180?

Or, 2x = 180?

Or, x = 90?

Therefore, angle AOB = 90? (Proved)


Now, In triangle ABD amd triangle BCD,

?AB=CB (Given)

angle ABO = angle CBO Or angle ABD = angle CBD

BD is common side

So triangle ABD congurent to CBD by SAS

Therefore, AD = CD (Proved)


Now, let us proceed to prove question (ii).

In triangle AOD and triangle COD,

AO = CO

OD is common side

AD = CD (we get)

Thus, ?AOD ? ?COD (By S-S-S rule of congruency).

?
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At the ending of the answer, it should be triangle AOD os congruent to triangle COD (proved). Thats a mathematical error which gets displayed in answers every time. Please ignore it.
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