Question

ABCD is a cyclic quadrilateral whose diagonals intersect at a point E.

If ∠ DBC = 70 °, ∠ BAC = 30°, find ∠ BCD.. Further, If AB = BC, find ∠ ECD

Answer 1

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Let's see some related topics :

⚫ Circle : The collection of all the points, which are at a fixed distance from a fixed point in a plane, is called a circle.


⚫ Radius : A line joining the centre to the Circumference of the circle, is called radius of a circle.


⚫ Secant : A line intersecting a circle at any two points, is called secant.


⚫ Diameter : A chord passing through the point of the circle, is called diameter. It is the longest chord.


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Answer 2

ABCD is a cyclic quadrilateral whose diagonal intersect at E.

∠CBD = ∠CAD (Angles in the same segment are equal)

∠CAD = 70°

∠BAD = ∠BAC + ∠CAD = 30° + 70° = 100°

∠BCD + ∠BAD = 180° (Opposite angles of a cyclic quadrilateral)

∠BCD + 100° = 180°

∠BCD = 80°

In ΔABC,

AB = BC (Given)

∴ ∠BCA = ∠CAB (Angles opposite to equal sides of a triangle are equal)

⇒ ∠BCA = 30°

But ∠BCD = 80°.

⇒ ∠BCA + ∠ACD = 80°

30° + ∠ACD = 80°

⇒ ∠ACD = 50°

⇒ ∠ECD = 50°