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In the Figure, Ab is the Chord of a Circle with Centre O and Doc is a Line Segment Such that Bc = Do. If ∠C = 20°, Find Angle Aod. - ICSE Class 10 - Mathematics

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ConceptChord Properties - a Straight Line Drawn from the Center of a Circle to Bisect a Chord Which is Not a Diameter is at Right Angles to the Chord

Question

In the figure, AB is the chord of a circle with centre O and DOC is a line segment such that BC = DO. If ∠C = 20°, find angle AOD.

Solution

Join OB,
In ΔOBC,
BC = OD = OB (Radii of the same circle)
∴ `∠`BOC = `∠`BCO = 20°
And Ext. `∠`ABO =  `∠`BCO + `∠`BOC
⇒ Ext..`∠`ABO = 20° + 20°  = 40° …… (i)
In ΔOAB,
OA = OB (radii of the same circle)
∴ `∠`OAB = `∠`OBA = 40° (from (i)
`∠`AOB = 180°  -  `∠`OAB - `∠`OBA
⇒ `∠`AOB  = 180°  - 40° - 40° = 100°
Since DOC is a straight line
∴ `∠`AOD+ `∠`AOB + `∠`BOC  = 180°
⇒ `∠`AOD + 100° + 20° = 180°
⇒ `∠`AOD = 180° - 120°
⇒ `∠`AOD = 60°

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Solution In the Figure, Ab is the Chord of a Circle with Centre O and Doc is a Line Segment Such that Bc = Do. If ∠C = 20°, Find Angle Aod. Concept: Chord Properties - a Straight Line Drawn from the Center of a Circle to Bisect a Chord Which is Not a Diameter is at Right Angles to the Chord.
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