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Theorem 7 and its Converse

Theorem 7: The sum of the either pair of the opposite angles of a cyclic quadrilateral is 180°. (Angles are supplementary).
Given: Let ABCD be a cyclic quadrilateral (Fig.)
To Proof: The sum of either pair of the opposite angles of a cyclic quadrilateral, is 180°. (Angles are supplementary).
Given: Let ABCD be a cyclic quadrilateral (Fig.)
To Proof:
A + C = 180° and B + D = 180°
Construction: Join OB and OD.

 

Proof
BOD = 2 BAD
BAD = BOD
similarly,
BCD = DOB
...
BAD + BCD= BOD + DOB
= (
BOD + DOB)
=
× 360°
= 180°
Similarly
B + D = 180°
 

Example :

If a side of a cyclic quadrilateral is produced, the exterior angle is equal to the interior opposite angle.

Solution :

 

In the cyclic quadrilateral ABCD,
B + D = 180° (sum of the opposite angles) -----(i)
But
ADC + ADE = 180° (Linear pair) ------------(ii)
from (i) and (ii)

...   B + ADC = ADC + ADE
or
B = ADE


Converse of Theorem 7
Converse :
If a pair of opposite angles of a quadrilateral is supplementary, the quadrilateral is cyclic.





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