Circle Theorems I (Copy)
1.
In a circle, AB is a diameter and C is a point on the circle.
Find ∠ACB.
2.
O is the centre of a circle. AB is a diameter, C is on the circle.
If ∠ACB = 90°, prove that AB is a diameter.
3.
O is the centre of the circle. Radius OA meets tangent PT at A.
Find ∠OAP.
4.
In a circle, ∠ABC at the circumference is 38°. A and C are joined to the centre O.
Find ∠AOC.
5.
In a circle, ∠AOC at the centre = 100°. A and C lie on circumference.
Find ∠ABC at the circumference.
6.
In a circle, points A, B, C, D lie on circumference.
If ∠ABC = 72°, find ∠ADC.
7.
In a circle, points P, Q, R, S form a cyclic quadrilateral.
If ∠PQR = 84°, find ∠PSR.
8.
In a cyclic quadrilateral, one angle is 115°.
Find the opposite angle.
9.
In a circle, ∠at centre = 120°. Find ∠at circumference standing on same arc.
10.
A tangent touches circle at A. Radius OA is drawn.
Find ∠OAT where T is point of tangent line.
11.
A triangle is drawn in a circle with base AB as diameter.
If point C lies on circumference, find ∠ACB.
12.
A tangent touches circle at P. A chord PQ is drawn.
If ∠between tangent and chord = 40°, find ∠in alternate segment.
13.
A tangent at A makes angle 65° with chord AB.
Find ∠ACB, where C is a point on opposite arc.
14.
In a circle, ∠AOB = 150°. A and B on circumference.
Find ∠ACB.
15.
ABCD is a cyclic quadrilateral.
If ∠A = 70°, ∠C = ?
16.
ABCD cyclic quadrilateral. ∠B = 95°.
Find ∠D.
17.
In a circle, ∠at circumference = 40°. Find angle at centre.
18.
In a circle, AB is diameter, C is point on circle.
If ∠CAB = 32°, find ∠CBA.
19.
In a cyclic quadrilateral, angles are in ratio 2:3:2:3.
Find all angles.
20.
A tangent touches circle at A. OA is radius.
Show why ∠OAT = 90°.
21.
In a circle, ∠at circumference = 56°. Find ∠at centre.
22.
In a circle, chord AB subtends 60° at circumference.
Find angle at centre.
23.
A tangent touches circle at P. PQ is chord.
If ∠between tangent and chord = 50°, find ∠opposite in alternate segment.
24.
A triangle is inscribed in a semicircle with diameter AB. Point C on semicircle.
Find ∠ACB.
25.
A cyclic quadrilateral has one angle 78°. Find its opposite angle.
26.
In a circle, ∠at circumference = 65°. Find ∠at centre.
27.
In a circle, ∠AOC = 140°. Find ∠ABC.
28.
In a circle, tangent touches at A. Radius OA drawn.
If OB is another radius, find ∠OAB.
29.
In a circle, tangent touches at P. Radius OP drawn.
Explain why ∠OPT = 90°.
30.
In a circle, a chord subtends 90° at circumference.
Find ∠at centre.
Answer Key and Explanations
1. ∠ACB = 90°.
Angle in semicircle = 90°.
2. If angle subtended by AB at circumference = 90°, then AB must be diameter.
Converse of angle in semicircle theorem.
3. ∠OAP = 90°.
Radius ⟂ tangent.
4. ∠AOC = 2 × ∠ABC = 2×38 = 76°.
Angle at centre twice angle at circumference.
5. ∠ABC = 100 ÷ 2 = 50°.
Angle at circumference half centre.
6. ∠ADC = ∠ABC = 72°.
Angles in same segment equal.
7. ∠PQR + ∠PSR = 180°.
∠PSR = 180 − 84 = 96°.
Cyclic quadrilateral opposite angles supplementary.
8. Opposite = 180 − 115 = 65°.
9. ∠at circumference = 120 ÷ 2 = 60°.
10. ∠OAT = 90°.
Radius perpendicular to tangent.
11. ∠ACB = 90°.
Angle in semicircle.
12. ∠in alternate segment = 40°.
Alternate segment theorem.
13. ∠ACB = 65°.
Alternate segment theorem.
14. ∠ACB = 150 ÷ 2 = 75°.
15. ∠C = 180 − 70 = 110°.
Cyclic quadrilaterals opposite angles supplementary.
16. ∠D = 180 − 95 = 85°.
17. ∠at centre = 2×40 = 80°.
18. ∠ACB = 90°.
∠CAB = 32°.
So ∠CBA = 180 − (90+32) = 58°.
19. Ratio 2:3:2:3.
Angles: 72°, 108°, 72°, 108°.
20. Tangent ⟂ radius at point of contact → 90°.
21. ∠at centre = 2×56 = 112°.
22. ∠at centre = 2×60 = 120°.
23. ∠in alternate segment = 50°.
24. ∠ACB = 90°.
Angle in semicircle.
25. Opposite = 180 − 78 = 102°.
26. ∠at centre = 2×65 = 130°.
27. ∠ABC = 140 ÷ 2 = 70°.
28. ∠OAB = 90°.
29. Tangent perpendicular to radius → ∠OPT = 90°.
30. ∠at centre = 2×90 = 180°.