Download presentation
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 = 2QPR
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
Similar presentations
© 2025 SlidePlayer.com. Inc.
All rights reserved.