A quadrilateral MNOP is inscribed in a circle. If MN is a diameter, what can you say about ∠MOP and ∠MNP Class 9

A quadrilateral MNOP is inscribed in a circle. If MN is a diameter, what can you say about ∠MOP and ∠MNP Class 9

Question 1.

A quadrilateral MNOP is inscribed in a circle. If MN is a diameter, what can you say about ∠MOP and ∠MNP? Explain your reasoning. Class 9

Solution:

MNOP is a cyclic quadrilateral, and MN is diameter of the circle.

∠MOP and ∠MNP are subtended on same chord MP.

Since angles in same segment are equal.

∴ ∠MOP = ∠MNP.

Question 2.

Let ABCD be a cyclic quadrilateral. Explain why the exterior angle at any vertex is equal to the interior opposite angle (e.g., ∠CDE = ∠ABC, where E is a point on the extension of side AD). Class 9

Solution:

ABCD is a cyclic quadrilateral

∴ ∠ADC + ∠ABC = 180°

... [Opposite angles are supplementary]

∴ ∠ABC = 180° - ∠ADC ...(i)

∠ADC + ∠CDE = 180° ...[linear pair]

∴ ∠CDE = 180° - ∠ADC ...(ii)

From (i) and (ii), we get

∠ABC = ∠CDE

[Note: The question has been modified]