Question
ABCD ia a cyclic quadrilateral in which BA and CD when produced meet in E and EA = ED. Prove that:
- AD || BC.
- EB = EC.
✓
Answer
Given ABCD is a cyclic quadrilateral in which EA = ED
To prove:
- AD || BC.
- EB = EC.
- Since EA = ED
Since, ABCD is a cyclic quadrilateral
Then, $\angle\text{ABC}+\angle\text{ADC}=180^\circ$
But $\angle\text{ABC}+\angle\text{EBC}=180^\circ$ [Linear pair of angles]
Then, $\angle\text{ADC}=\angle\text{EBC}\dots(2)$
Compare equations (1) and (2)
$\angle\text{EAD}=\angle\text{EBC}\dots(3)$
Since, corresponding angles are equal
Then, BC || AD
- From equation (3)
Similarly $\angle\text{EDA}=\angle\text{ECB}\dots(4)$
Compare equations (1)(3) and (4)
$\angle\text{EBC}=\angle\text{ECB}$
$\Rightarrow \text{EB}=\text{EC}$ [Opposite angles to equal sides]
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