Question

Given : Two circles intersect each other at C and D. Line AB is their common tangent.
To prove: $\angle ACB+\angle ADB={180}^{\circ }$

Answer 1

Given,

two circles intersect at each other at $C$ & $D$

$\overline{AB}$ is their common tangent.

Also, given

to prove that

$\angle ACB+\angle ADB={180}^o$

$\angle CBA=\angle CDB------\left(1\right)$

$\angle CAB=\angle CDA-------\left(2\right)$

$\angle CDB+\angle CDA=\angle CBA+\angle CAB\to eq\left(1\right)$

$\angle CBA+\angle CAB+\angle ACB={180}^o$

$\angle CBA+\angle CAB={180}^o-\angle ACB-----eq\left(2\right)$

from $\left(1\right)$ & $\left(2\right)$

$\angle CDB+\angle CDA={180}^o-\angle ACB$

$\angle CDB+\angle CDA=\angle ADB$

$\overline{={180}^o-\angle ACB-\angle ADB}$

$\angle ADB={180}^o-\angle ACB$

$\therefore \angle ADB+\angle ACB={180}^o$.