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Prove that the tangents drawn at the end points of a chord of a circle make equal angles with the chord.
Given: Two tangents drawn at the end points of a chord of a circle.
To do: The tangents make equal angles with the chord.
Solution:
![](/assets/questions/media/148618-33689-1607066997.png)
Need to prove that $\angle BAP\ =\angle \ ABP$
$AB$ is the chord.
We know that $OA = OB\ ( radius)$
$\angle OBP=\angle OAP=90^{o}$
Join $OP$ and
$OP=OP$
By SAS congruency
$\vartriangle OBP\cong \vartriangle OAP$
$\therefore \ BP=AP$
Angles opposite to equal sides are equal.
$\therefore \angle BAP=\angle ABP$
Hence proved $\angle BAP=\angle ABP$
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