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Vinci & his teamUltimate Revision Handbook Deductive Geometry

HKDSE Maths (Compulsory)

HKDSE Maths (Compulsory) ) (

Ultimate Revision Handbook ()

Deductive Geometry

Includes more than 80 reasons essential for HKDSE Maths (Compulsory) deductive geometry () NOT FOR SALE Copyright by Vinci Mak P.1

Vinci & his teamUltimate Revision Handbook Deductive Geometry

HKDSE Maths (Compulsory)

1. Angles of Triangle ( ) 1.1 In any triangle, A + B + C = 180 Reason : [ sum of ] / [ ] 1.2 In any triangle,

ABC + BAC = ACDReason : [ext. of ] / [ ]

2. Angles and lines ( ) 2.1 If AO, BO, CO and DO intersect at O, then AOBOCO DO O

a + b + c + d = 360Reason : [ s at a pt.] / [] 2.2 If AB and CD intersect at O, then AB CD O

a=bReason :

&

x=y

[vert. opp. s] / []

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Vinci & his teamUltimate Revision Handbook Deductive Geometry

HKDSE Maths (Compulsory)

3. Polygons ( ) 3.1 Sum of interior angles of a n-sided polygon (n) = (n 2) 180

Reason : [ sum of polygon] / [] 3.2 Sum of exterior angles of convex polygon = 360 Reason : [ext. of polygon] / []

4. Straight lines ( ) 4.1 If AOB is a straight line, then AOB a + b = 180 Reason : [adj. s on st. line] / [] 4.2 If a + b = 180 , then a + b = 180

is a straight line.AOB Reason : [adj. s supp.] / []

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Vinci & his teamUltimate Revision Handbook Deductive Geometry

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5. Parallel lines ( ) 5.1 If PQ // RS, then ( PQ // RS) (a) a = b Reason : [corr. s, PQ // RS] / [PQ // RS] (b) b = c Reason : [alt. s, PQ // RS] / [PQ // RS] (c) b + d = 180 Reason : [int. s, PQ // RS] / [PQ // RS] 5.2 PQ // RS if either PQ // RS (a) a = b Reason : [corr. s eq.] / [] (b) b = c Reason : [alt. s eq.] / [] (c) b + d = 180 Reason : [int. s supp.] / []

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Vinci & his teamUltimate Revision Handbook Deductive Geometry

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6. Isosceles triangles ( ) 6.1 If AB = AC, then AB = AC B = C Reason : [base s, isos. ] / [ ] 6.2 If B = C , then B = C AB = AC Reason : [side opp.. eq. s] / [] 6.3 ABC is an isosceles triangle with AB = AC if ABC AB = AC B = C Reason : [base s eq.] / []

7. Equilateral triangle () 7. If AB = BC = CA, then AB = BC = CA A = B = C = 60 Reason : [Property of equi. ] / [ ]B C A

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Vinci & his teamUltimate Revision Handbook Deductive Geometry

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8. Right-angled triangle ( ) 8.1 If C = 90 , then C = 90 a 2 + b2 = c 2 Reason : [Pyth. theorem] / [] 8.2 If a 2 + b 2 = c 2 , then a 2 + b 2 = c 2 C = 90 / ABC is a right-angled . Reason : [Converse of Pyth. theorem] []

9. Congruent triangles ( ) 9.1 If ABC XYZ , then AB = XY BC = YZ AC = XZ Reason : [corr. sides, s] / [ ] A = X B = Y C = Z Reason : [corr. s, s] / [ ]

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9.2 ABC XYZ if AB = XY BC = YZ CA = ZX Reason : [S.S.S.]

9.3 ABC XYZ if B = Y C = ZBC = YZ Reason : [A.S.A.]

9.4 ABC XYZ if B = Y C = ZBA = YX Reason : [A.A.S.]

9.5 ABC XYZ if A = X BA = YXAC = XZ Reason : [S.A.S.]

9.6 ABC XYZ if ABC XYZ BA = YX AC = XZ C = Z = 90

Reason : [R.H.S.]

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10. Similar triangles ( ) 10.1 ABC ~ XYZ if A = X B = Y C = Z Reason : [A.A.A.] / [] 10.2 ABC ~ XYZ if

AB BC CA = = XY YZ ZXReason : [3 sides prop.] / [] 10.3 ABC ~ XYZ if

AB CA = & A = X XY ZXReason : [ratio of 2 sides, inc. ] [] 10.4 If ABC ~ XYZ , then

AB BC CA = = XY YZ ZX Reason :[corr. sides, ~ s] / [ ~ ]

A = X B = Y C = ZReason : [corr. s, ~ s] / [ ~ ]

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11. Quadrilaterals 11.1 Trapezium A quadrilateral having 1 pair of parallel sides Properties : 1. A + D = 180 2. B + C = 180 Reason : [Property of trapezium] / []

11.2 Parallelogram A quadrilateral having 2 pairs of parallel sides 1. AB // DC 2. AD // BC Reason : [Property of //gram] / []

Properties :11.2A If ABCD is a parallelogram, then 1. AB = CD 2. AD = BC Reason : [opp. sides of //gram] / [] 11.2B If ABCD is a parallelogram, then 1. A = C 2. B = D Reason : [opp. s of //gram] / [] 11.2C If ABCD is a parallelogram, then 1. AO = OC 2. BO = OD Reason : [diags. of //gram] / [] P.9

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11.3 Rhombus A parallelogram having 4 equal sides 4 Properties : 1. All properties of //gram 1. 2. Diagonals bisect each interior angle 2. 3. Diagonals to each other 3. Reason : [Property of rhombus] / []

11.4 Rectangle A parallelogram having 4 right angles 4 Properties : 1. All properties of //gram 1. 2. Diagonals are equal in length. 2. Reason : [Property of rectangle] / []

11.5 Square A parallelogram having 4 right angles & 4 equal sides 44 Properties : 1. All properties of //gram 1. 2. Diagonals are equal in length. 2. 3. Diagonals to each other 3. 4. Angle between diagonal and side = 45 4. 45

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Reason : [Property of square] / [

]

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11.6 Kite A quadrilateral having 2 pairs of adjacent equal sides 2 Properties : Diagonals to each other Reason : [Property of kite] / []

12. Proofs of parallelogram ( ) 12.1 ABCD is a parallelogram if ABCD 1. AB = DC 2. AD = BC Reason : [opp. sides eq.] / [] 12.2 ABCD is a parallelogram if ABCD 1. A = C 2. B = D Reason : [opp. s eq.] / [] 12.3 ABCD is a parallelogram if ABCD 1. AO = OC 2. BO = OD Reason : [diags. bisect each other] / [] 12.4 ABCD is a parallelogram if ABCD 1. AB = DC 2. AB // DC Reason :[opp. sides eq. and //] / [

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]

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HKDSE Maths (Compulsory)

13. Mid-point theorem ( ) 13.1 If AM = MB and AN = NC, then AM = MBAN = NC 1. MN // BC

1 BC 2 Reason : [Mid-pt. thm.] / []2. MN =

13.2 If MN // BC and MN =

1 BC , then 2

MN // BC MN =

1 BC 2

1. AM = MB 2. AN = NC Reason : [Converse of mid-pt. thm.] / []

14. Intercept theorem ( ) 14.1 If AB // CD // EF, then AB // CD // EF BD AC = DF CE Reason : [Intercept thm.] / [] 14.2 If BD AC BD AC = , then / = DF CE DF CE AB // CD // EF Reason : [Converse of intercept thm.] / []

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HKDSE Maths (Compulsory)

15. Relationships between sides and angles of triangle () 15.1. Triangle inequality In any triangle a+b>c a+c>b b+c>a Reason : [Triangle inequality] / [] 15.2. Greater angle, greater side A. If A > B > C , then A > B > C a>b>c Reason : [Greater , greater side] / [ ] B. If a > b > c, then a > b > c A > B > C Reason : [Greater side, greater ] / [ ]

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16. Points of concurrence in triangle ( ) 16.1 Centroid If point G is the centroid of the triangle, then G CD = DB Reason : [Centroid of ] / [ ] 16.2 Incentre If point I is the incentre of the triangle, then I I CAD = BAD Reason : [Incentre of ] / [ ] 16.3 Orthocentre If point R is the orthocentre of the triangle, then R C AD CB D Reason : [Orthocentre of ] / [ ] 16.4 Circumcentre If point C is the circumcentre of the triangle, then C D CE DB & DE = EB E Reason : [Circumcentre of ] / [ ] B A B R A C D B A C D B A

G

C

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17. Chords of a circle ( ) 17.1. A perpendicular line from the centre of a circle to a chord bisects the chord.

If ON AB, then AN = NB. ON ABAN = NB Reason : [ from centre to chord bisects chord] []

17.2. The line joining the centre of a circle and the mid-point of a chord is perpendicular to the chord.

If AM = MB, then OM AB. AM = MBOM AB Reason : [line joining centre and mid-pt. of chord chord] / [] 17.3 The centre of a circle lies on the perpendicular bisector of a chord. Reason : [ bisector of chord passes through centre] []

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17.4 If two chords of a circle are equal in length, then they are equidistant from the centre.

If AB = CD, then OM = ON. AB = CDOM = ON Reason : [eq. chords equidistant from centre] []

17.5 If two chords of a circle are equidistant from the centre, then their lengths are equal.

If OM = ON, then AB = CD. OM = ONAB = CD Reason : [chords equidistant from centre eq.] []

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(Maths)(Physics) (Applied Maths) A A

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18. Angles in a circle ( ) 18.1 The angle at the centre of a circle subtended by an arc is twice the angle at the circumference subtended by the same arc.

x = 2y. Reason : [ at centre twice at []ce

]

Vinci

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18.2 If AB is a diameter and P is any point on the circumference except A and B, then AB P A B

APB = 90 . Reason : [ in semi-circle] / []

18.3 If APB = 90 , then AB is a diameter. APB = 90 AB Reason : [Converse of in semi-circle] []

18.4 Angles in the same segment of a circle are equal. Reason : x=y [ s in the same segment] / []

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19. Relationships among Arcs, Chords & Angles ( )19.1 Equal angles at the centre of a circle (or equal circles) stand on equal arcs and equal chords.

If a = b, then / a = b

1. AB = CD [eq. s, eq. arcs] / [] 2. AB = CD [eq. s, eq. chords] / []

19.2 Equal chords on a circle stand on equal arcs and equal angles.

If AB = CD, then / AB = CDa=b [eq. chords, eq. s] / []

AB = CD

[eq. chords, eq. arcs] / []

19.3 Equal arcs in a circle (or equal circles) subtend equal angles at the centre.

If AB = CD , then / AB = CD a=b [eq. arcs, eq. s] / [] AB = CD [eq. arcs, eq. chords] / []

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19.4 In a circle (or equal circles), the lengths of arcs are proportional to the sizes of angles at the centre subtended by the arcs.

AB : BC = m : n

Reason : [arcs prop. to s at centre] []

19.5 In a circle (or equal circles), the lengths of arcs are proportional to the sizes of angles at the circumference.

AB : BC = x : y

Reason : [arcs prop. to s at ce] []

Vinci 2010 1A4B3C 2010 (Nov) IGCSE Maths 1 Edexcel A* 2 Cambridge A

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20. Cyclic Quadrilateral ( ) 20.1 The opposite angles of a cyclic quadrilateral are supplementary. x + y = 180

Reason : [opp. s, cyclic quad.] []

20.2 For a cyclic quadrilateral, an exterior angle is equal to its interior opposite angle.

a=b

Reason : [ext. s, cyclic quad.] []

Vinci

F.1 - F.3 Maths F.1 - F.2 Integrated Science F.3 Physics & Chemistry HKDSE Maths (Compulsory, M1, M2) GCSE Maths GCSE Physics IGCSE Maths IGCSE Physics

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21. Tests for Concyclic Points ( ) 21.1 If p = q, then A, B, Q and P are concyclic. p = q ABC D Reason : [Converse of s in the same segment] []

21.2 If A + C = 180 or B + D = 180 , then A, B, C and D are concyclic. A + C = 180 ABC D

Reason: [opp. s supp.] / []

21.3 If p = q, then A, B, C and D are concyclic. p = q ABC D Reason : [ext. = int. opp. ] []

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22. Properties of Tangents of a circle ( ) 22.1 If PQ is a tangent to the circle at T, then PQ T OT PQ.

Reason : [tangent radius] / [ ] Conversely, if OT PQ, then PQ is a tangent to the circle at T. OT PQ PQ T Reason : [converse of tangent radius] [ ]

22.2 Perpendicular to a tangent at its point of contact passes through the centre of the circle.

22.3 If TP and TQ are two tangents to a circle at P and Q respectively, then T P

Q TP TQ(a) TP = TQ; (b) TOP = TOQ; (c) PTO = QTO. Reason : [tangents from ext. pt.] / [tangent properties] [] / []

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23. Angle in alternate segment ( ) 23.1 If PQ is a tangent to the circle at A, then PQ A AC x=y

Reason : [ in alt. segment] / [] 23.2 If x = y, then PQ is the tangent to the circle at A. x = y PQ A Reason : [Converse of in alt. segment] []

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