1. Angle ABC is subtended by arc AC. Name the theorem:

2. Line segment AD is a tangent at point A. Name the theorem relating angle BAD and angle ACD:

3. Diameter EF creates a right angle at point G on the circumference. Name the theorem:

1. Given angle ACB = 40°, calculate angle ADB:

2. If angle PQR = 70° and PR is a diameter, find angle PSR:

3. In a cyclic quadrilateral ABCD, angle ABC = 85° and angle CDA = 95°. Find angle DAB:

1. Prove that the angle subtended by an arc at the centre is twice the angle subtended at the circumference.

2. Prove that opposite angles of a cyclic quadrilateral sum to 180°.

A circular park has a path that forms a chord of the circle. The path is 60 metres long and subtends an angle of 90° at the centre of the park. Calculate the radius of the park.

1. The angle in a semicircle theorem vs. the tangent-segment theorem.

2. The alternate segment theorem vs. the cyclic quadrilateral theorem.

1. In a circle, the radius is 10 cm. A chord is 16 cm long. Find the distance from the centre of the circle to the chord.

2. Two tangents are drawn from a point outside the circle to points A and B on the circle. If angle AOB = 120°, find the angle between the tangents.

1. If a tangent and a secant are drawn from a point outside the circle, explain the relationship between the segments formed.

2. Calculate the length of the tangent segment if the secant segment is 15 cm and the external segment is 9 cm.

1. Prove that a triangle inscribed in a circle with one side as the diameter is a right triangle using coordinate geometry.

2. Verify the theorem that states the perpendicular from the centre of a circle to a chord bisects the chord using coordinates.