# Q : 8       Bisectors of angles A, B and C of a triangle ABC intersect its circumcircle at D, E and                 F respectively. Prove that the angles of the triangle DEF are    $\small 90^{\circ}-\frac{1}{2}C$,  $\small 90^{\circ}-\frac{1}{2}B$ and $\small 90^{\circ}-\frac{1}{2}A$

S seema garhwal

Given :   Bisectors of angles A, B and C of a triangle ABC intersect its circumcircle at D, E and F respectively.

To prove :  the angles of the triangle DEF are    $\small 90^{\circ}-\frac{1}{2}C$,    $\small 90^{\circ}-\frac{1}{2}B$ and $\small 90^{\circ}-\frac{1}{2}A$

Proof :

$\angle$1 and $\angle$3 are angles in same segment.therefore,

$\angle$1 = $\angle$3 ................1(angles in same segment are equal )

and    $\angle$2 = $\angle$4 ..................2

$\angle$1+$\angle$2=$\angle$3+$\angle$4

$\Rightarrow \angle D=\frac{1}{2}\angle B+\frac{1}{2}\angle C$,

$\Rightarrow \angle D=\frac{1}{2}(\angle B+\angle C)$

$\Rightarrow \angle D=\frac{1}{2}(180 \degree+\angle C)$

and  $\Rightarrow \angle D=\frac{1}{2}(180 \degree-\angle A)$

$\Rightarrow \angle D=90 \degree-\frac{1}{2}\angle A$

Similarly,  $\Rightarrow \angle E=90 \degree-\frac{1}{2}\angle B$     and    $\angle F=90 \degree-\frac{1}{2}\angle C$

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