T'AT is a tangent drawn to the circle with center O If angle BAC = 50 degree ,find - Maths - Circles

Question

T'AT is a tangent drawn to the circle with center O If angle BAC =
50 degree ,find angle BAT and angle BCA Also find angle DAB+angle
DCB

Manbar Singh · Accepted Answer

We know that tangent to a circle is always ⊥ to its radius at the point of contact
Now, T'AT is a tangent to the circle at point A
So, OA⊥T'AT  or ∠OAT = 90°Now, ∠OAT = 90°⇒∠CAT = 90°⇒∠CAB + ∠BAT = 90°⇒50° + ∠BAT = 90°⇒∠BAT =40°Now, ∠ABC = 90°   Angle in a semicircle is a right angleIn ∆ABC,∠ABC + ∠BAC + ∠BCA = 180°   Angle sum property⇒90° + 50° + ∠BCA = 180°⇒140° + ∠BCA = 180°⇒∠BCA = 40°Now, ABCD is a cyclic quadrilateral, then∠DAB + ∠DCB = 180°   Opposite angles of cyclic quadrilateral are supplementary

T'AT is a tangent drawn to the circle with center O . If angle BAC = 50 degree ,find angle BAT and angle BCA .Also find angle DAB+angle DCB