The sum of pairs of opposite angles of a cyclic quadrilateral is:

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3600

900

1800

600

answer is C.

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Detailed Solution

We know that the cyclic quadrilateral is the quadrilateral whose all four corners lie on the circumference of a circle.
Let us assume a cyclic quadrilateral ABCD that has AC and BD as its diagonals intersecting at point O.
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We know the property that the angles in the same segment are always equal ……………………………..(1)
From the above diagram, for chord CD,
∠DAC = ∠DBC …………………………….….(2)
Similarly, for chord BC,
∠CDB = ∠CAB …………………………….….(3)
Similarly, for chord AB,
∠ADB = ∠ACB …………………………….….(4)
Similarly, for chord AD,
∠ABD = ∠ACD …………………………….….(5)
We know that the sum of all angles of a quadrilateral is 3600 …………………………………(6)
⇒ ∠DAC + ∠DBC + ∠CDB + ∠CAB + ∠ADB + ∠ACB + ∠ABD + ∠ACD = 3600 ……………………………………….(7)
From equation (2), equation (3), equation (4), equation (5), and equation (7), we get
⇒ ∠DAC + ∠DBC + ∠CDB + ∠CAB + ∠ADB + ∠ACB + ∠ABD + ∠ACD = 3600 ⇒ ∠DAC + ∠DAC + ∠CAB + ∠CAB + ∠ACB + ∠ACB + ∠ACD + ∠ACD = 3600
⇒ 2 (∠DAC + ∠CAB + ∠ACB + ∠ACD) = 3600
⇒ ∠DAC + ∠CAB + ∠ACB + ∠ACD = 1800 ……………………………………………(8)
From the above figure, we can see that
∠DAB = ∠DAC + ∠CAB …………………………………………..(9)
∠BCD = ∠ACB + ∠ACD …………………………………………….(10)
Now, from equation (8), equation (9), and equation (10), we get
⇒ ∠DAB + ∠BCD = 1800 …………………………………………….(11)
Also, from the above figure, we can see that ∠DAB and ∠BCD are opposite angles ……………………………(12)
From equation (11) and equation (12), we can say that the sum of the opposite angles is equal to 1800.
Therefore, the correct option is (3).

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