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Geometry

This resource provides a detailed exploration of various geometric proofs and computations. It covers topics such as tangent lines, isosceles triangles, cyclic quadrilaterals, and the relationship between angles at the center and circumference of a circle. Each proof is broken down into logical steps, accompanied by geometric reasons and diagrammatic explanations. Students will learn to calculate angles in polygons, demonstrate properties of circles, and understand the significance of parallel lines and transversals. Ideal for students looking to strengthen their geometry skills.

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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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