Question

In the following figure, what can be concluded from the below given conditions.

Given: AB = CD and AC = DE.

• A. AC = BD, $\angle$AOC = $\angle$BOD
• B. $\angle$AOC = $\angle$DOE, AC = DE
• C. AD = CE, $\angle$AOD = $\angle$COE.
• D. All of the above.

Answer 1

The correct option is D

All of the above.

Given that chords AB and CD are equal.

Therefore, the angles subtended by them at the centre are also equal.

$\angle AOB=\angle COD$

We can write $\angle AOB=\angle AOC+\angle COB$------ (1)

Similarly we can write $\angle COD=\angle COB+\angle BOD$ ------(2)

From (1) and (2)

Equating $\angle AOB=\angle COD$

$\angle$AOC + $\angle$COB = $\angle$COB + $\angle$BOD

Therefore, $\angle$AOC = $\angle$BOD

Therefore using converse of theorem, we get AC = BD.

Also it is given that AC = DE

Therefore, AC = DE = BD

Hence the angle subtended by these chords will be equal.

$\angle$AOC = $\angle$DOE

It is given that AB = CD

If we add equal quantities on both sides then it does not makes any difference.

Therefore if we add BD to AB and add DE to CD (BD = DE)

AB + BD = CD + DE

AD = CE

Consequently, $\angle$AOD = $\angle$COE.