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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
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