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

• Angle XBC = 90

• Reason:

Question 1

• Angle XBC = 90

• Reason:

• Radius is perpendicular to tangent

• (Rad.tang.)

Question 1

• Angle CBA = ?

• Reason:Adjacent angles on a line add up to 180

Question 1

• Angle CBA = 20

• Reason:Adjacent angles on a line add up to 180

Question 1

• Angle CAB = 20

• Reason:

Question 1

• Angle CAB = 20

• Reason: Base angles of an isosceles triangle

• (Base s isos.∆)

Question 1

• Hence AXB = 50

• Reason sum of the angles in a triangle is 180

• ( sum ∆)

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°.

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.)

Question 2

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

Question 3

• Find the value of k

Question 3

• k = 107• (cyclic quad.)

Question 4

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

Question 4

CAB = BCA• (Base s isos ∆)

Question 4

EDB = • (opposite angles of

parallelogram)

Question 4

EDB = EAB• (opposite angles of

parallelogram)

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

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

Question 5

• AD is parallel to BC• 1. Find the sizes of the

marked angles.

Question 5

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

Question 5

• 2. Give a geometrical reason why PQ is parallel to RS.

• Co-int. s sum to 180• Or• Alt. s are equal

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

Question 6

PRO = a • (base angles isosceles

triangle) SOR = 2a • (ext. ∆)

Question 6

• Similarly SOQ = 2b

QOR = 2a + 2b QOR = 2(a + b) QOR = 2QPR

Question 7

• AD, AC and BD are chords of the larger circle.

• AD is a diameter of the smaller circle.

Question 7

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

Question 7

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

• p = 43• (s same arc)

Question 7

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

• q = 90• ( in a semi-circle)

Question 7

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

• r = 47• (ext. ∆)

Question 7

• Is E the centre of the larger circle?

Question 7

• Is E the centre of the larger circle?

• No because base angles ACD and BDC are not equal.

Question 8

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

Question 8

• Sione correctly calculated that x = 56

• Write down the geometric reason for this answer.

Question 8

• Sione correctly calculated that x = 56

• Write down the geometric reason for this answer.

• Cyclic quad.

Question 8

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

Question 8

• y = 90• ( in a semi-circle)

Question 8

• z = 28• (base s isos. ∆)

Question 9

• You are asked to prove triangle BCF is isosceles.

• Fill in the blanks to complete the proof.

B C

F

Question 9

BCF = 38° .• (alt. s // lines)B C

F

Question 9

BFC = 38° .• (adj ’s on st. line

add to 180)

B C

F