Question

If one side of a cyclic quadrilateral is produced, then the exterior angle is equal to the interior angle. Prove it.

Answer 1

Given: Side DC of cyclic quadrilateral ABCD is produced and point E is taken on it.

To Prove: ∠BCE = ∠BAD

Proof:

∠BCE+∠BCD = 180⁰ ___ (1)

(Angles in a linear pair)

∠BAD+∠BCD=180⁰ ___ (2)

(Opposite angles of a cyclic quadrilateral)

From (1) and (2),

∠BCE = ∠BAD

Hence an exterior angle of cyclic quadrilateral is congruent to the angle opposite to its adjacent interior angle.

Answer 2

Given : 

• A cyclic quadrilateral ABCD one of whose side AB is produced to E. 

Prove that : 

• ∠CBE = ∠ADC 

Proof:

• In cyclic quadrilateral ABCD,

We know,

• ⟼ ∠ABC + ∠ADC = 180° ----(1)

[Opposite angles of cyclic quadrilateral.]

• ⟼ ∠ABC + ∠CBE = 180° ------(2)

[Linear Pair angles.]

From (1) and (2), equate both equations we get

• ⟼ ∠ABC + ∠ADC = ∠ABC + ∠CBE  

This implies,

• ⟼ ∠ADC = ∠CBE

Additional Information -

• The definition states that a quadrilateral which is circumscribed in a circle is called a cyclic quadrilateral. It means that all the four vertices of quadrilateral lie in the circumference of the circle.

• The sum of the opposite angles of a cyclic quadrilateral is supplementary. 

Let ∠A, ∠B, ∠C and ∠D are the four angles of an inscribed quadrilateral. Then,

• ∠A + ∠C = 180°

• ∠B + ∠D = 180°