Angles at the centre and circumference

1. Minor central angle AOB is ${84}^{\circ }$. C lies on the major arc AB. Find $\angle ACB$.

   2 marks
2. $\angle ACB={37}^{\circ }$ subtends minor arc AB. Find the corresponding minor central angle.

   2 marks
3. Minor central angle AOB is ${120}^{\circ }$. D lies on the minor arc AB. Find $\angle ADB$.

   2 marks
4. C and D lie on the same arc of chord AB. $\angle ACB={46}^{\circ }$. Find $\angle ADB$.

   2 marks
5. AB is a diameter of a circle and C is another point on the circle. $\angle CAB={32}^{\circ }$. Find $\angle ABC$.

   2 marks
6. An inscribed angle subtends a chord and is ${90}^{\circ }$. Explain why the chord is a diameter.

   3 marks

Equivalent correct methods earn the corresponding marks. Include units where asked.

1. Minor central angle AOB is ${84}^{\circ }$. C lies on the major arc AB. Find $\angle ACB$.

   2 marks

   Same minor arc:$\frac{84}{2}=42$.

   ${42}^{\circ }$
2. $\angle ACB={37}^{\circ }$ subtends minor arc AB. Find the corresponding minor central angle.

   2 marks

   Double 37.

   ${74}^{\circ }$
3. Minor central angle AOB is ${120}^{\circ }$. D lies on the minor arc AB. Find $\angle ADB$.

   2 marks

   Opposite arc is major:$360-120=240$.

   $\frac{240}{2}=120$.

   ${120}^{\circ }$
4. C and D lie on the same arc of chord AB. $\angle ACB={46}^{\circ }$. Find $\angle ADB$.

   2 marks

   They are angles in the same segment.

   ${46}^{\circ }$
5. AB is a diameter of a circle and C is another point on the circle. $\angle CAB={32}^{\circ }$. Find $\angle ABC$.

   2 marks

   $\angle ACB={90}^{\circ }$.

   $180-90-32=58$.

   ${58}^{\circ }$
6. An inscribed angle subtends a chord and is ${90}^{\circ }$. Explain why the chord is a diameter.

   3 marks

   The corresponding central angle is ${180}^{\circ }$.

   The endpoints are opposite ends of a diameter.

   Its intercepted arc is a semicircle.