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Geometry

Geometry. Proofs. Question 1. In this diagram (which is not drawn to scale), C is the centre of the circle, and XY is a tangent to the circle. The angle ABY equals 70°. Question 1. Fill in the gaps in the table below to find, in 4 logical steps, which angle equals 50°. Question 1.

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Geometry

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  1. Geometry Proofs

  2. Question 1 • In this diagram (which is not drawn to scale), C is the centre of the circle, and XY is a tangent to the circle. • The angle ABY equals 70°.

  3. Question 1 • Fill in the gaps in the table below to find, in 4 logical steps, which angle equals 50°.

  4. Question 1 • Angle XBC = 90 • Reason:

  5. Question 1 • Angle XBC = 90 • Reason: • Radius is perpendicular to tangent • (Rad.tang.)

  6. Question 1 • Angle CBA = ? • Reason:Adjacent angles on a line add up to 180

  7. Question 1 • Angle CBA = 20 • Reason:Adjacent angles on a line add up to 180

  8. Question 1 • Angle CAB = 20 • Reason:

  9. Question 1 • Angle CAB = 20 • Reason: Base angles of an isosceles triangle • (Base s isos.∆)

  10. Question 1 • Hence  AXB = 50 • Reason sum of the angles in a triangle is 180 • ( sum ∆)

  11. Question 2 • The Southern Cross is shown on the New Zealand flag by 4 regular five-pointed stars. • The diagram shows a sketch of a regular five-pointed star. • When drawn accurately, the shaded region will be a regular pentagon, and the angle PRT will equal 108°.

  12. Question 2 • Calculate, with geometric reasons, the size of angle PQR in a regular 5-pointed star (You should show three steps of calculation, each with a geometric reason.)

  13. Question 2 • PRQ = 72 • (adj. s on a line) • RPQ = 72 • (base s isos ∆) • PQR = 36 • ( sum ∆)

  14. Question 3 • Find the value of k

  15. Question 3 • k = 107 • (cyclic quad.)

  16. Question 4 • Complete the following statements to prove that the points B, D, C and E are concyclic

  17. Question 4 • CAB = BCA • (Base s isos ∆)

  18. Question 4 • EDB = • (opposite angles of parallelogram)

  19. Question 4 • EDB = EAB • (opposite angles of parallelogram)

  20. Question 4 • Therefore B, D, C and E are concyclic points because the • opposite angles of a quadrilateral are supplementary. • exterior angle of a quadrilateral equals interior opposite angle. • equal angles are subtended on the same side of a line segment

  21. Question 4 • Therefore B, D, C and E are concyclic points because the • equal angles are subtended on the same side of a line segment

  22. Question 5 • AD is parallel to BC • 1. Find the sizes of the marked angles.

  23. Question 5 • x = 56 • (adj. s on a line) • y = 33 • (alt. s // lines)

  24. Question 5 • 2. Give a geometrical reason why PQ is parallel to RS. • Co-int. s sum to 180 • Or • Alt. s are equal

  25. Question 6 • You are asked to prove "the angle at the centre is twice the angle at the circumference". • Fill in the blanks to complete the proof that • QOR = 2 x QPR

  26. Question 6 • PRO = a • (base angles isosceles triangle) • SOR = 2a • (ext.  ∆)

  27. Question 6 • Similarly SOQ = 2b • QOR = 2a + 2b • QOR = 2(a + b) • QOR = 2QPR

  28. Question 7 • AD, AC and BD are chords of the larger circle. • AD is a diameter of the smaller circle.

  29. Question 7 • Write down the size of the angles marked p, q and r.

  30. Question 7 • Write down the size of the angles marked p, q and r. • p = 43 • (s same arc)

  31. Question 7 • Write down the size of the angles marked p, q and r. • q = 90 • ( in a semi-circle)

  32. Question 7 • Write down the size of the angles marked p, q and r. • r = 47 • (ext. ∆)

  33. Question 7 • Is E the centre of the larger circle?

  34. Question 7 • Is E the centre of the larger circle? • No because base angles ACD and BDC are not equal.

  35. Question 8 • In the diagram 0 is the centre of the circle. BC = CD.

  36. Question 8 • Sione correctly calculated that x = 56 • Write down the geometric reason for this answer.

  37. Question 8 • Sione correctly calculated that x = 56 • Write down the geometric reason for this answer. • Cyclic quad.

  38. Question 8 • Write down the sizes of the other marked angles giving reasons for your answers.

  39. Question 8 • y = 90 • ( in a semi-circle)

  40. Question 8 • z = 28 • (base s isos. ∆)

  41. Question 9 • You are asked to prove triangle BCF is isosceles. • Fill in the blanks to complete the proof. B C F

  42. Question 9 • BCF = 38° . • (alt. s // lines) B C F

  43. Question 9 • BFC = 38° . • (adj ’s on st. line add to 180) B C F

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