2. Angle Theorems
(4) The angle subtended by an arc (or chord) at the centre is double the
angle subtended by the arc (or chord) at the circumference.
3. Angle Theorems
(4) The angle subtended by an arc (or chord) at the centre is double the
angle subtended by the arc (or chord) at the circumference.
B
O
A C
4. Angle Theorems
(4) The angle subtended by an arc (or chord) at the centre is double the
angle subtended by the arc (or chord) at the circumference.
AOC 2ABC at centre, twice at circumference on same arc
B
O
A C
5. Angle Theorems
(4) The angle subtended by an arc (or chord) at the centre is double the
angle subtended by the arc (or chord) at the circumference.
AOC 2ABC at centre, twice at circumference on same arc
B Prove : AOC 2ABC
O
A C
6. Angle Theorems
(4) The angle subtended by an arc (or chord) at the centre is double the
angle subtended by the arc (or chord) at the circumference.
AOC 2ABC at centre, twice at circumference on same arc
B Prove : AOC 2ABC
Proof: Join BO and produce to X
O
A X
C
7. Angle Theorems
(4) The angle subtended by an arc (or chord) at the centre is double the
angle subtended by the arc (or chord) at the circumference.
AOC 2ABC at centre, twice at circumference on same arc
B Prove : AOC 2ABC
Proof: Join BO and produce to X
O AOB is isosceles OA OB, radii
A X
C
8. Angle Theorems
(4) The angle subtended by an arc (or chord) at the centre is double the
angle subtended by the arc (or chord) at the circumference.
AOC 2ABC at centre, twice at circumference on same arc
B Prove : AOC 2ABC
Proof: Join BO and produce to X
O AOB is isosceles OA OB, radii
OBA OAB base' s isosceles
A X
C
9. Angle Theorems
(4) The angle subtended by an arc (or chord) at the centre is double the
angle subtended by the arc (or chord) at the circumference.
AOC 2ABC at centre, twice at circumference on same arc
B Prove : AOC 2ABC
Proof: Join BO and produce to X
O AOB is isosceles OA OB, radii
OBA OAB base' s isosceles
A X AOX OBA OAB exterior OAB
C
10. Angle Theorems
(4) The angle subtended by an arc (or chord) at the centre is double the
angle subtended by the arc (or chord) at the circumference.
AOC 2ABC at centre, twice at circumference on same arc
B Prove : AOC 2ABC
Proof: Join BO and produce to X
O AOB is isosceles OA OB, radii
OBA OAB base' s isosceles
A X AOX OBA OAB exterior OAB
C
AOX 2OBA
11. Angle Theorems
(4) The angle subtended by an arc (or chord) at the centre is double the
angle subtended by the arc (or chord) at the circumference.
AOC 2ABC at centre, twice at circumference on same arc
B Prove : AOC 2ABC
Proof: Join BO and produce to X
O AOB is isosceles OA OB, radii
OBA OAB base' s isosceles
A X AOX OBA OAB exterior OAB
C
AOX 2OBA
COX 2OBC by similar method
12. Angle Theorems
(4) The angle subtended by an arc (or chord) at the centre is double the
angle subtended by the arc (or chord) at the circumference.
AOC 2ABC at centre, twice at circumference on same arc
B Prove : AOC 2ABC
Proof: Join BO and produce to X
O AOB is isosceles OA OB, radii
OBA OAB base' s isosceles
A X AOX OBA OAB exterior OAB
C
AOX 2OBA
COX 2OBC by similar method
AOC 2ABC
14. (5a) The angle in a semicircle is a right angle.
A
C
O
B
15. (5a) The angle in a semicircle is a right angle.
ACB 90 in a semicircle
A
C
O
B
16. (5a) The angle in a semicircle is a right angle.
ACB 90 in a semicircle
A
C Data : AOB is diameter
O
B
17. (5a) The angle in a semicircle is a right angle.
ACB 90 in a semicircle
A
C Data : AOB is diameter
Prove : ACB 90
O
B
18. (5a) The angle in a semicircle is a right angle.
ACB 90 in a semicircle
A
C Data : AOB is diameter
Prove : ACB 90
O Proof: AOB 180 straight
B
19. (5a) The angle in a semicircle is a right angle.
ACB 90 in a semicircle
A
C Data : AOB is diameter
Prove : ACB 90
O Proof: AOB 180 straight
at centre twiceat
AOB 2ACB
B circumference on same arc
20. (5a) The angle in a semicircle is a right angle.
ACB 90 in a semicircle
A
C Data : AOB is diameter
Prove : ACB 90
O Proof: AOB 180 straight
at centre twiceat
AOB 2ACB
B circumference on same arc
ACB 90
21. (5a) The angle in a semicircle is a right angle.
ACB 90 in a semicircle
A
C Data : AOB is diameter
Prove : ACB 90
O Proof: AOB 180 straight
at centre twiceat
AOB 2ACB
B circumference on same arc
ACB 90
Exercise 9B; 1 ace etc, 2, 6, 8ac, 9ab, 10ac, 11ac, 12, 13