Question

In the figure O is the centre of the circle. The tangents at B and D intersect each other at point P. If AB is parallel to CD and $\angle ABC={55}^o$ Find (i) $\angle BOD$ (ii) $\angle BPD$

Answer 1

In the figure AB || CD ......... (Given)
$\therefore \angle ABC=\angle BCD$ .......... (Alternate angles)
But $\angle ABC={55}^o$ ......... (Given)
$\therefore \angle BCD={55}^o$
In the figure $\angle BOD=2\angle BCD$ ........ (Central angle formed by an arc is twice the inscribed angle formed by the same arc)
$\therefore \angle BOD=2\times {55}^o={110}^o$

$\therefore \angle BOD={110}^o$ ....... $\left(i\right)$
But $\angle BOD+\angle BPD={180}^o$ ........ [Opposite angles of a quadrilateral are supplementary]
$\therefore \angle BPD={180}^o-\angle BOD={180}^o-{110}^o={70}^o$

$\therefore \angle BPD={70}^o$ ......... $\left(ii\right)$