Math topic中文

Angle properties

angle properties of circles:

  • angle in a semicircle is a right angle
  • angle between tangent and radius of a circle is a right angle
  • angle at the centre is twice the angle at the circumference
  • angles in the same segment are equal
  • angles in opposite segments are supplementary

Practice questions

  1. 1A circle with centre O has a tangent XY at D. ∠ODC = 70°, ∠BCD = 120° and ∠ODA = 15°. Find ∠CDY. Find ∠DAB. Find ∠OBA. T…
  2. 2In the diagram, BA and BC are tangents to the circle with centre O. BO meets the circle at D, OA = 10 cm and BC = 28 cm.…
  3. 3The diagram shows a circle PQRST, with centre O. PV and RV are tangents to the circle and U is the intersection of PS an…
  4. 4A, B and C are points on the circle with centre O. Q and R are points on the circle with centre P. CSR and BSQ are tange…
  5. 5下列关于圆的叙述正确的有（ ） ①圆内接四边形的对角互补； ②相等的圆周角所对的弧相等； ③正多边形内切圆的半径与正多边形的半径相等； ④圆内接平行四边形是矩形． A. 1个 B. 2个 C. 3个 D. 4个
  6. 6⊙O的弦AB的长等于半径，那么弦AB所对的圆周角等于 度．
  7. 7In the diagram, AB is a tangent to the circle with centre O, ∠CAB = 58° and CDE is a straight line. Find, stating your r…
  8. 8The diagram shows a semi-circle BAC, with diameter BC = 34 cm. AD and BCD are straight lines. Given that cos⁡∠ABC=817\cos\angle ABC = \dfrac{8}{17}…
  9. 9In the diagram, P, Q, R and S are points on a circle. PR and QS meet at T. Show that triangle PTS and triangle QTR are s…
  10. 10The points A, B, E and D lie on the circle with centre O. BEC is a straight line and BF is a tangent to the circle at po…
  11. 11同弧所对的圆周角等于圆心角的 ．
  12. 12PQRS are points on the circumference of a circle, centre O. Angle OQR = 50°, angle PSR = 73° and obtuse angle SOQ = 156°…
  13. 13In the diagram, A, B, C and D are four points on a circle, centre O. Angle ABO=x∘ABO = x^\circ and angle ADO=y∘ADO = y^\circ. Fi…
  14. 14A, B and C are points on the circle, centre O. DE is a tangent to the circle at A. Angle AOC = 4x∘4x^\circ and angle ACB…
  15. 15B, C, D, E and F are points on the circle with centre O. AE is tangent to the circle and ABC is a straight line. ∠BAE=38∘\angle BAE = 38^\circ…
  16. 16O is the centre of the circle passing through B, E, D and C. Angle DOC=84∘DOC = 84^\circ and angle CBO=54∘CBO = 54^\circ. PQPQ is…
  17. 17在△ABC\triangle ABC中，点OO是不与AA、BB、CC重合的点，且满足OA=OB=OCOA=OB=OC，若∠BOC=140°∠BOC=140°，则∠BAC=∠BAC= ___ .
  18. 18PR is a diameter of the large circle, centre O. QS is a diameter of the small circle, centre O. PQ and RS are tangents t…
  19. 19AC is a diameter of a circle with centre O. AC=2rAC = \sqrt{2}r and AB=BCAB = BC. COB is a semi-circular arc, centre P. What p…
  20. 20A logo, as shown below, has a circle with centre O and radius 2 cm, an equilateral triangle ABC of side 5 cm and an isos…
  21. 21A, B, C, D and E are points on the circle, centre O. QP and QR are tangents to the circle at B and E respectively. AC in…
  22. 22In the diagram, OAB is a sector of a circle with centre O. Given that OA = 12 cm, ∠AOB = θ\theta radians, OC = 8 cm and…
  23. 23In the diagram, BDK is a straight line and AB = AD. Angle ABD = 36°, angle CBD = 49° and angle CDK = 121°. Explain why i…
  24. 24A, B and C are points on the circle centre O and AB = BC. P is the midpoint of chord AB and Q is the midpoint of chord B…
  25. 25In the diagram, OAPB is a sector of circle with centre O and radius 9 cm. Angle AOB = 1.4 radians. C is the centre of th…
  26. 26In the diagram, A, B C and D lie on a circle, centre O. PQ and PR are tangents to the circle at A and D respectively. It…
  27. 27In the diagram, AC is a diameter of the circle ABCD. AB and DC are produced to meet at E. Angle AED = 20° and angle CAD…
  28. 28The diagram shows three points A(-16, 0), B(0, 12) and C(8, 2). A circle is drawn with centre C such that AB is a tangen…
  29. 29In the diagram, ABAB is the diameter of the circle. AED and ABC are straight lines. AE=12AE = 12 cm and EB=16EB = 16 cm. ∠ADC\angle ADC…
  30. 30A chock is a block placed against a wheel or rounded object, to prevent it from moving. Jenny had to come up with a prot…
  31. 31The diagram shows a circle, with centre O, passing through A, B, C, D and E. CG and FG are tangents to the circle at poi…
  32. 32The figure below shows a circle with centre O. Points A, B and C lie on the circle. Angle ABC = 100°. Find angle OAC, gi…
  33. 33A, B, C and D are points on the circle with centre O and radius 5 cm. AE is a tangent to the circle at A and OE is a str…
  34. 34In the diagram, the points A, B, C and D lie on a circle, centre O. ∠ADC = 118° and ∠BAO = 32°. Find, giving reasons for…
  35. 35In the diagram, ABTC is a cyclic quadrilateral. QTP is a tangent to the circle and ∠BAT=2∠CAT\angle BAT = 2\angle CAT. The lines…
  36. 36The diagram shows a circle BDEFG, centre O and diameter BE. The line AC is a tangent to the circle at B and ∠BOD=30∘\angle BOD = 30^\circ…
  37. 37The diagram shows two concentric circles with centre O. PQ is the diameter of the larger circle and RS is the diameter o…
  38. 38In the diagram, O is the centre of the circle. DOEFG and AEB are straight lines and GA is a tangent to the circle at A.…
  39. 39In the diagram, O is the centre of the circle through A, B, C, D and TB is the tangent at B. AC is the diameter of the c…
  40. 40A, B, C and D are points on a circle. Line PCQ is a tangent to the circle. Line QE bisects angle AEC and is parallel to…
  41. 41若⊙O的一条弧所对的圆周角为60°，则这条弧所对的圆心角是 ．
  42. 42The diagram, not drawn to scale, shows two concentric circles, centre O. AB and AD are tangents to the smaller circle at…
  43. 43The diagram shows a circle with points A, B, D and E on the circle and a point C outside the circle. A, B, D and E are p…
  44. 44A, B, D and E are points on the circle such that AD = CD and angle BCD = 28°. Explain with geometrical reasons, why the…
  45. 45In the diagram, ABC and AED are straight lines and BE is parallel to CD. Show that triangle ABE is similar to triangle A…
  46. 46In the figure, AT and CT are tangents to the circle at A and C respectively. AB produced meets TC at P. Given that ∠APC…
  47. 47Triangle ABC is such that AB = 16 cm, BC = 63 cm, AC = 65 cm. Do points A, B and C lie on the circumference of a circle?…
  48. 48P, R and Q are points on the circle with centre, O. PT and QT are tangents to the circle at P and Q respectively. ∠PTQ =…
  49. 49设双曲线CC：x2a2−y2b2=1(a>0,b>0)\dfrac{{{x}^{2}}}{{{a}^{2}}}-\dfrac{{{y}^{2}}}{{{b}^{2}}}=1\left(a>0,b>0\right)的左、右焦点分别为F1{{F}_{1}}，F2{{F}_{2}}…
  50. 50In the figure, circle ABCF and circle CDEF intersect at C and F. AFD and BFE are straight lines and BC is parallel to AD…
  51. 51The diagram shows a quadrilateral ABCDABCD made up of two triangles. AB=20 cm, AD=12 cm, CD=32 cm.AB = 20\ \text{cm},\ AD = 12\ \text{cm},\ CD = 32\ \text{cm}.…
  52. 52In the diagram, not drawn to scale, A, B, C, D and E are points on the circle. CPE and BPD are straight lines. It is giv…
  53. 53In the diagram, A, B, C and D are points on the circumference of a circle with centre O. AB is parallel to DO and ∠AOD=67∘\angle AOD = 67^\circ…
  54. 54In the diagram, O is the centre of the circle and ABCD is a cyclic quadrilateral. Angle AOB=∠ADC=70∘AOB = \angle ADC = 70^\circ. E…
  55. 55In the diagram ABC is a straight line, BC=10 cmBC = 10\ \text{cm}, CD=8 cmCD = 8\ \text{cm} and BD=6 cmBD = 6\ \text{cm}. Explain why a…
  56. 56The lines ABC and CDE are tangents to the circle at B and D respectively. F, G and H are points on the circle and the ce…
  57. 57In the diagram, P, Q, R, S and T are points on the circle with centre O. VX is a tangent to the circle at P. Line SR is…
  58. 58In the diagram, A, B, C and D are points on the circle with centre O. AG and BG are tangents to the circle. GAE, FAD and…