Question

Points B, D, E and F lie on a circle.
ABC is the tangent to the circle at B.
Find the size of angle ABD.
You must give a reason for each stage of your working.

Answer 1

Given :- Points B, D, E and F lie on a circle. ABC is the tangent to the circle at B.

To Find :- ∠ABD = ?

Solution :-

→ ∠BDF = 40° (given)

so,

→ ∠CBF = ∠BDF { Alternate segment theorem. }

→ ∠CBF = 40° --------- Eqn.(1)

now,

→ ∠DEF = 100° (given)

so,

→ ∠DBF = 180° - ∠DEF = 180° - 100° = 80° { sum of opposite angles of cyclic quadrilateral is equal to 180°.} ------ Eqn.(2)

therefore,

→ ∠ABD + ∠DBF + ∠CBF = 180° { AC is a straight line.}

→ ∠ABD + 80° + 40° = 180°

→ ∠ABD = 180° - 120°

→ ∠ABD = 60° (Ans.)

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