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# In the Given Figure, ∠Bad = 65°, ∠Abd = 70° and ∠Bdc = 45°. Find: (I) ∠Bcd (Ii) ∠Acb Hence, Show that Ac is a Diameter - ICSE Class 10 - Mathematics

ConceptArc and Chord Properties - Angles in the Same Segment of a Circle Are Equal (Without Proof)

#### Question

In the given figure, ∠BAD = 65°, ∠ABD = 70° and ∠BDC = 45°. Find:
(i) ∠BCD (ii) ∠ACB
Hence, show that AC is a diameter #### Solution ∠BCD = 180° - ∠BAD = 180° - 65° = 115°
(pair of opposite angles in a cyclic quadrilateral are supplementary)

(ii) By angle sum property of ∆ABD,
ADB = 180° - 65° - 70° = 45°
Again, ∠ACB = ∠ADB = 45°
(Angle in the same segment)
∴ ∠ADC = ∠ADB + ∠BDC = 45° + 45° = 90°
Hence, AC is a semicircle.
(since angle in a semicircle is a right angle)

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Solution In the Given Figure, ∠Bad = 65°, ∠Abd = 70° and ∠Bdc = 45°. Find: (I) ∠Bcd (Ii) ∠Acb Hence, Show that Ac is a Diameter Concept: Arc and Chord Properties - Angles in the Same Segment of a Circle Are Equal (Without Proof).
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