Circles [NEW] — Maths STD 9 — Question

1. AD || BC.
2. EB = EC.

1. Since EA = ED

Then, $\angle\text{EAD}=\angle\text{EDA}\dots(1)$ [Oppo. angles to equal sides]

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

2. From equation (3)

$\angle\text{EAD}=\angle\text{EBC}\dots(3)$

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]