Question

Show that the tangents at the extremities of a chords of a circle makes equal angles with the chord.

Answer 1

Let $PQ$ be the chord of a circle with center $O$.

Let $AP$ and $AQ$ be the tangents at points $P$ and $Q$ respectively.

Let us assume that both the tangents meet at point $A$.

Join points $O,P$. let $OA$ meets $PQ$ at $R$

Here we have to prove that $\angle APR=\angle AQR$

Consider, $\Delta APR$ and $\Delta AQR$

$AP=AQ$ (Tangents drawn from an internal point to a circle are equal0

$\angle PAR=\angle QAR$

$AR=AR$ {common side}

$\therefore \Delta APR\cong \Delta AQR$ [SAS congruence criterion]

Hence,

$\angle APR=\angle AQR\left[CPCT\right]$