Question

In the figure, seg $AB$ and seg $AD$ are the chords of the circle. $C$ is any point on the tangent of the circle at the point $A$. If $m\left(arcAPB\right)={80}^o$ and $m\angle BAD={30}^o$. Find $m\angle BAC$ and $m\left(arcBQD\right)$

Answer 1

$m\left(arcAPB\right)={80}^o$
$\therefore m\angle BAC=\frac{1}{2}\times m\left(arcAPB\right)$ (Tangent secant theorem)
$=\frac{1}{2}\times {80}^o={40}^o$
$\angle BAD={30}^o$ ....(Given)
$\therefore \angle BAD=\frac{1}{2}\times m\left(arcBQD\right)$ ....(inscribed angle theorem)
$\therefore {30}^o=\frac{1}{2}\times m\left(arcBQD\right)$
$\therefore m\left(arcBQD\right)=30\times 2={60}^o$
$\therefore m\angle BAC={40}^o$ and $m\left(arcBQD\right)={60}^o$