Question

In the figure, O is the centre of the circle. What is the value of x?
(i) 125°(ii) 105°(iii) 95°(iv) 85°
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Answer 1

Answer :

(ii) 105°

Concept to be used :

• Cyclic quadrilateral : A quadrilateral whose all the four vertices lies on the circumference of a circle is called a cyclic quadrilateral .

• The sum of opposite angles of a cyclic quadrilateral is 180° .

• In another words , both the pairs of opposite angles of a cyclic quadrilateral are supplementary .

• Measure of angles subtended to any point on the circumference of a circle from an arc is equal to half of the angle subtended at the center by the same arc .

Solution :

• Given : ∠AOC = 150°
• To find : ∠ABC = x = ?
• Construction : Take a point D anywhere on the major arc AC , then join the points A and C to the point D .

We know that ,

Measure of angles subtended to any point on the circumference of a circle from an arc is equal to half of the angle subtended at the center by the same arc .

Thus ,

→ ∠ADC = ∠AOC/2

→ ∠ADC = 150°/2

→ ∠ADC = 75°

Clearly ,

Quadilateral ABCD is a cyclic quadrilateral , thus sum of its opposite angles will be 180° .

→ ∠ADC + ∠ABC = 180°

→ 75° + x = 180°

→ x = 180° - 75°

→ x = 105°

Hence , x = 105° .