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# In the Adjoining Figure, O is the Centre of the Circle and Ab is a Tangent to It at Point B. ∠Bdc = 65° Find ∠Bao. - ICSE Class 10 - Mathematics

ConceptTangent Properties - If a Line Touches a Circle and from the Point of Contact, a Chord is Drawn, the Angles Between the Tangent and the Chord Are Respectively Equal to the Angles in the Corresponding Alternate Segments

#### Question

In the adjoining figure, O is the centre of the circle and AB is a tangent to it at point B. ∠BDC = 65° Find ∠BAO.

#### Solution

AB is a straight line.
∴ ∠ADE +  ∠BDE = 180°
⇒ ∠ADE  + 65° = 180°
⇒ ∠ADE = 115°          ………..(i)
AB i.e. DB is tangent to the circle at point B and BC is the diameter.

∴∠DB = 90°
In ΔBDC,
∠DBC +  ∠BDC + ∠DCB = 180°
⇒ 90° +  65° +  ∠DCB  = 180°
⇒ ∠ DCB = 25°
Now, OE = OC (radii of the same circle)
∴ ∠DCB or ∠OCE =  ∠OEC = 25°
Also,
∠OEC = ∠DEC =  25°

(vertically opposite angles)
∠ADE + ∠DEA + ∠DAE = 180°
From (i) and (ii)
115° + 25°  + ∠ DAE  = 180°
⇒ ∠DAE or ∠BAO  = 180° - 140° =40°
∴∠BAO = 40°

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Solution In the Adjoining Figure, O is the Centre of the Circle and Ab is a Tangent to It at Point B. ∠Bdc = 65° Find ∠Bao. Concept: Tangent Properties - If a Line Touches a Circle and from the Point of Contact, a Chord is Drawn, the Angles Between the Tangent and the Chord Are Respectively Equal to the Angles in the Corresponding Alternate Segments.
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