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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

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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)
In  ΔADE,
`∠`ADE + `∠`DEA + `∠`DAE = 180°
From (i) and (ii)
115° + 25°  + `∠` DAE  = 180°
⇒ `∠`DAE or `∠`BAO  = 180° - 140° =40°
∴`∠`BAO = 40°

  Is there an error in this question or solution?

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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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