Worksheet Chapter 6: Angles and Tangents of Circles

1. True or False: A radius meeting a tangent line at the point of contact forms a right angle (90°).

2. True or False: The sum of any two adjacent interior angles in a cyclic quadrilateral is always equal to 180°.

3. A circle with centre O has points A, B, C, and D on its circumference forming a cyclic quadrilateral. Given that reflex angle AOC = 230° and angle ABD = 40°. Calculate: (a) Angle ABC. (b) Angle ADC. (c) Angle DBC.

4. In a circle, a line PQR is a tangent touching the circle at Q. A triangle QST is inscribed inside the circle such that angle PQS = 64° and angle TQR = 48°. Calculate: (a) Angle STQ. (b) Angle SQT.

5. In a circle with centre O, triangle ABC is inscribed such that AC is the diameter. If angle BAC = 32°, find the value of angle BCA.

6. ABCD is a cyclic quadrilateral. Given that angle DAB = (2x + 10)° and angle BCD = (3x - 15)°, find the value of x.

7. In the diagram, TP and TQ are tangents to a circle with centre O at points P and Q respectively. Given that angle PTQ = 46° and radius OP = 9 cm. Calculate: (a) Angle POQ at the centre. (b) Angle OPQ. (c) The length of line OT (to 2 decimal places).

8. Two tangents TA and TB are drawn from external point T to a circle with centre O and radius 7 cm. If the distance OT = 25 cm, calculate: (a) The length of tangent TA. (b) The area of quadrilateral OATB.

9. Fill in the blank: The angle subtended by an arc at the centre of a circle is ________ the angle subtended by the same arc at the circumference.

10. In a circle with centre O, arc AB subtends an angle of 130° at the centre. Calculate the angle subtended by arc AB at the circumference.