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