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 = 2QPR

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