Question 31: is a diameter of a circle, as shown in the figure. and are straight lines. Find (i) (ii)  (iii) 

Answer:

(i) (angles in the same segment of the circle subtended by the same chord)

(ii)

(angle in a semi circle subtended by the diameter)

(iii)

Question 32: In the given figure, is bisector of  and is a cyclic quadrilateral. Prove that: .

Answer:

is a cyclic quadrilateral

(cyclic quadrilateral)

Also (angles in the same segment of the circle subtended by the same chord)

Question 33: In the figure, is the center of the circle, . Calculate  and .

Answer:

Question 34: In the given figure, and are the centers of two intersecting circles intersecting at and . is a straight line. Calculate the numerical value of .

Answer:

is a straight line

Question 35: In the figure given below, two circles intersect at and . The center of the smaller circle is and lines on the circumference of the larger circle. Given . Find in terms of the value of (i) Obtuse (ii)  (iii) . Give reasons.

Answer:

(i)

(ii) is a cyclic quadrilateral)

(iii) (angles in the same segment of the circle subtended by the same chord)

Question 36: In the given figure is the cent of the circle and . Calculate and .

Answer:

is a cyclic quadrilateral

Question 37: In the given figure, is the center of the circle, is a parallelogram and is a straight line. Prove that .

Answer:

(angle subtended at the center is twice subtended on the circumference by the same chord)

(alternate angles)

is a parallelogram

(opposite angles in a parallelogram are equal)

 

Question 38: is a cyclic quadrilateral in which is parallel to and is a diameter of the circle. Given ; calculate: (i)  (ii) .

Answer:

(angles in the same segment of the circle subtended by the same chord)

(angle subtended by the diameter on a semi circle)

(alternate angles)

Question 39: In the given figure is the diameter of the circle.  Chord and . Calculate (i)  (ii) .

Answer:

(i) (angle in the semi circle)

(ii)

(alternate angles)

is a cyclic quadrilateral

Question 40: The sides and of a cyclic quadrilateral are produced to meet at , the sides and are produced to meet at . If  and find (i)  (ii) 

Answer:

(vertically opposite angles)

(cyclic quadrilateral)