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In the Following Figure, Pq is the Tangent to the Circle at A, Db is the Diameter and O is the Centre of the Circle. If ∠Adb = 30° and ∠Cbd = 60°, Calculate: (I) ∠Qab, (Ii) ∠Pad, (Iii) ∠Cdb, - 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 following figure, PQ is the tangent to the circle at A, DB is the diameter and O is the centre of the circle. If ∠ADB = 30° and ∠CBD = 60°, Calculate:
(i) ∠QAB, (ii) ∠PAD, (iii) ∠CDB, 

Solution

i) PAQ is a tangent and AB is the chord.
`∠`QAB = `∠`ADB = 30° (angles in the alternate segment)
ii) OA = OD (radii of the same circle)
∴ `∠`OAD  =`∠`ODA = 30°
But, OA ⊥ PQ
∴ `∠`PAD = `∠` OAP  - `∠` OAD = 90°  - 30° = 60°
iii) BD is the diameter.
∴ `∠`BCD = 90 (angle in a semi – circle)
Now in ΔBCD,
`∠` CDB + `∠`CBD + `∠`BCD  =180°
⇒ `∠` CDB  + 60° + 90°  =180° 
⇒ `∠`CDB =180° - 150° = 30° 

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Solution In the Following Figure, Pq is the Tangent to the Circle at A, Db is the Diameter and O is the Centre of the Circle. If ∠Adb = 30° and ∠Cbd = 60°, Calculate: (I) ∠Qab, (Ii) ∠Pad, (Iii) ∠Cdb, 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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