Geometry Proofs

Question 1In 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 1Fill in the gaps in the table below to find, in 4 logical steps, which angle equals 50.

Question 1Angle XBC = 90Reason:

Question 1Angle XBC = 90Reason:Radius is perpendicular to tangent(Rad.tang.)

Question 1Angle CBA = ?Reason:Adjacent angles on a line add up to 180

Question 1Angle CBA = 20Reason:Adjacent angles on a line add up to 180

Question 1Angle CAB = 20Reason:

Question 1Angle CAB = 20Reason: Base angles of an isosceles triangle(Base s isos.)

Question 1Hence AXB = 50Reason sum of the angles in a triangle is 180( sum )

Question 2The 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 2Calculate, 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 2PRQ = 72 (adj. s on a line)RPQ = 72(base s isos )PQR = 36( sum )

Question 3Find the value of k

Question 3k = 107(cyclic quad.)

Question 4Complete the following statements to prove that the points B, D, C and E are concyclic

Question 4CAB = BCA(Base s isos )

Question 4EDB = (opposite angles of parallelogram)

Question 4EDB = EAB(opposite angles of parallelogram)

Question 4Therefore 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 4Therefore B, D, C and E are concyclic points because the

equal angles are subtended on the same side of a line segment

Question 5AD is parallel to BC1. Find the sizes of the marked angles.

Question 5x = 56(adj. s on a line)y = 33(alt. s // lines)

Question 52. Give a geometrical reason why PQ is parallel to RS.Co-int. s sum to 180OrAlt. s are equal

Question 6You 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 6PRO = a (base angles isosceles triangle) SOR = 2a (ext. )

Question 6Similarly SOQ = 2b

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

Question 7AD, AC and BD are chords of the larger circle. AD is a diameter of the smaller circle.

Question 7Write down the size of the angles marked p, q and r.

Question 7Write down the size of the angles marked p, q and r.p = 43(s same arc)

Question 7Write down the size of the angles marked p, q and r.q = 90( in a semi-circle)

Question 7Write down the size of the angles marked p, q and r.r = 47(ext. )

Question 7Is E the centre of the larger circle?

Question 7Is E the centre of the larger circle?No because base angles ACD and BDC are not equal.

Question 8In the diagram 0 is the centre of the circle. BC = CD.

Question 8Sione correctly calculated that x = 56Write down the geometric reason for this answer.

Question 8Sione correctly calculated that x = 56Write down the geometric reason for this answer.Cyclic quad.

Question 8Write down the sizes of the other marked angles giving reasons for your answers.

Question 8y = 90( in a semi-circle)

Question 8z = 28(base s isos. )

Question 9You are asked to prove triangle BCF is isosceles. Fill in the blanks to complete the proof.

BCF

Question 9BCF = 38 .(alt. s // lines)

BCF

Question 9BFC = 38 .(adj s on st. line add to 180)

BCF