Question

In the given figure if $\angle ACE={43}^{\circ }$ and $\angle CAF={62}^{\circ }$, find the value of $a,b$ and $c$.

Answer 1

Now, $\angle ACE={43}^o$ and $\angle CAF={62}^o$[given]

In $\Delta AEC$

$\therefore \angle ACE+\angle CAE+\angle AEC={180}^o$

${43}^o+{62}^o+\angle AEC={180}^o$

${105}^o+\angle AEC={180}^o$

$\angle AEC={180}^o-{105}^o={75}^o$

Now, $\angle ABD+\angle AED={180}^o$

[Opposite angles of a cyclic quadrilateral and $\angle AED=\angle AEC$]

$a+{75}^o={180}^o$

$a={180}^o-{75}^o$

$a={105}^o$

$\angle EDF=\angle BAF$

$\therefore c={62}^o$ [Angles in the alternate segments]

In $\Delta BAF$,

$a+{62}^o+b={180}^o$

${105}^o+{62}^o+b={180}^o$

${167}^o+b={180}^o$

$\Rightarrow b={180}^o-{167}^o={13}^o$

Hence, $a={105}^o,b={13}^o$ and $c={62}^o$.