Ex 10.5, 6 - ABCD is a cyclic quadrilateral whose diagonals - Cyclic quadrilaterals

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Ex 10.5, 6 ABCD is a cyclic quadrilateral whose diagonals intersect at a point E. If ∠DBC = 70°, ∠BAC is 30°, find ∠BCD. Further, if AB = BC, find ∠ECD. For segment DCBAD ∠ CBD & ∠ CAD are in the same segment So, they must be equal ∴ ∠ CAD = ∠ CBD ∴ ∠ CAD = 70° Also, ∠BAD = ∠BAC + ∠CAD ∠BAD = 30° + 70° ∠BAD = 100° Now, ABCD is a cyclic quadrilateral ∠BCD + ∠BAD = 180° ∠BCD + 100° = 180° ∠BCD = 180° – 100° ∠ BCD = 80° Also, if AB = BC, we need to find ∠ ECD In ∆ ABC, AB = BC So, ∠ BCA = ∠ BAC ∴ ∠ BCA = 30° Also, ∠ BCD = ∠ BCA + ∠ ECD 80° = 30° + ∠ ECD ∠ ECD = 80° – 30° ∠ ECD = 50°

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