In figure 3.58, seg RS is a diameter of the circle with centre O. Point T lies in the exterior of the circle. Prove that ∠ RTS is an acute angle.

Solve any two sub-question:

If P(-2,4), Q(4,8), R(10, 5) and S(4, 1) are the vertices of a quadrilateral, show that it is a parallelogram.

Solve Question:

Find the angle between two radii at the center of the circle as shown in the figure, Lines PA and PB are tangents to the circle at other ends of the radii and ∠APR = 110°.

Solve sub-questions:

Prove that “the opposite angles of cyclic quadrilateral are supplementary”

Solve any the three sub-questions:

ABCD is a cyclic quadrilateral. M (arc ABC) = 230°. Find ∠ABC, ∠CDA and ∠CBE.

Solve Question:

Prove that “the opposite angles of a cyclic quadrilateral are supplementary”

In figure 3.91, line PR touches the circle at point Q. Answer the following questions with the help of the figure.

(1) What is the sum of ∠ TAQ and∠ TSQ?

(2) Find the angles which are congruent to ∠ AQP.

(3) Which angles are congruent to ∠QTS ?

(4) ∠ TAS = 65°, find the measure of ∠TQS and arc TS.

(5) If ∠AQP = 42°and ∠SQR = 58° find measure of ∠ATS.

Prove that any three points on a circle cannot be collinear.

In figure 3.99, seg MN is a chord of a circle with centre O. MN = 25,L is a point on chord MN such that ML = 9 and d(O,L) = 5.

Find the radius of the circle.

In figure 3.58, seg RS is a diameter of the circle with centre O. Point T lies in the exterior of the circle. Prove that ∠ RTS is an acute angle.

In figure 3.61, chords AC and DE intersect at B. If ∠ ABE = 108°, m(arc AE) = 95°, find m(arc DC).