In the given figure, O is the center of a circle and two tangents CA, CB are drawn on the circle from a point C lying outside the circle. Prove that $∠ A O B$ and $∠ A C B$ are supplementary.

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Solution

In MQNP,
$m ∠ C A O = 90^{0}$ (Radius is perpendicular to the tangent)
$m ∠ C B O = 90^{0}$ (Radius is perpendicular to the tangent)
$∴ m ∠ A O B + m ∠ A C B = 360^{0} - m ∠ C A O - m ∠ C B O = 360^{0} - 90^{0} - 90^{0} = 180^{0}$
Hence, angle AOB and angle ACB are supplementary.

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In the given figure, O is the center of a circle and two tangents CA, CB are drawn on the circle from a point C lying outside the circle. Prove that ∠AOB and ∠ACB are supplementary.

In MQNP,m∠CAO=900 (Radius is perpendicular to the tangent)m∠CBO=900 (Radius is perpendicular to the tangent)∴m∠AOB+m∠ACB=3600−m∠CAO−m∠CBO=3600−900−900=1800 Hence, angle AOB and angle ACB are supplementary.

In the given figure, O is the center of a circle and two tangents CA, CB are drawn on the circle from a point C lying outside the circle. Prove that $∠ A O B$ and $∠ A C B$ are supplementary.

In MQNP,
$m ∠ C A O = 90^{0}$ (Radius is perpendicular to the tangent)
$m ∠ C B O = 90^{0}$ (Radius is perpendicular to the tangent)
$∴ m ∠ A O B + m ∠ A C B = 360^{0} - m ∠ C A O - m ∠ C B O = 360^{0} - 90^{0} - 90^{0} = 180^{0}$
Hence, angle AOB and angle ACB are supplementary.