HKDSE Mathematics Ronald Hui Tak Sun Secondary School

14 September 2015Ronald HUI Missing Homework SHW1-A1 SHW1-A1 10, 24 10, 24 SHW1-B1 SHW1-B1 10, 14 10, 14 SHW1-C1 SHW1-C1 Today!!!! Today!!!! Summer Holiday Homework Summer Holiday Homework 25 Sep (Fri) 25 Sep (Fri)

14 September 2015Ronald HUI

14 September 2015Ronald HUI

14 September 2015Ronald HUI

14 September 2015Ronald HUI

Book 5A Chapter 1 Cyclic Quadrilaterals

ABCD is called a cyclic quadrilateral. A B C D  A and  C  a pair of opposite angles In the figure, all the vertices of quadrilateral ABCD lie on a circle.  i.e. ABCD is inscribed in the circle.  a pair of opposite angles  B and  D Opposite Angles of a Cyclic Quadrilateral

Let me show you that the opposite angles of a cyclic quadrilateral are supplementary, i.e. their sum is equal to 180 . Is there any relationship between a pair of opposite angles of a cyclic quadrilateral? A B C D

O y x A B C D Join OB and OD. Let ∠ BOD = x and reflex ∠ BOD = y. x = 2 ∠ BAD and y = 2 ∠ BCD  at centre twice  at  ce ∵ x + y = 360   s at a pt. ∴ 2 ∠ BAD + 2 ∠ BCD = 360  ∠ BAD + ∠ BCD = 180  i.e. ∠ A + ∠ C = 180  By similar arguments, ∠ ABC + ∠ ADC = 180  i.e. ∠ B + ∠ D = 180 

Theorem 1.18 The opposite angles of a cyclic quadrilateral are supplementary.  A +  C = 180   B +  D = 180  Abbreviation: opp.  s, cyclic quad. A B C D

A B C 50  x  ADC +  ABC = 180  D 85  opp.  s, cyclic quad. (x + 50  ) + 85  = 180  Example: Find x in the figure. x = 45 

Follow-up question Find x in the figure. B 28  A D C x ∵ BC = BA (given) (  sum of △ ) (base  s, isos. △ ) ∴ In △ ABC, 124  ∠ ABC = 124 

Follow-up question Find x in the figure. ∵ BC = BA (given) (  sum of △ ) (base  s, isos. △ ) ∴ In △ ABC, B 28  A D C x ∠ ABC = 124  x + ∠ ABC = 180  (opp. ∠ s, cyclic quad.) x  = 180  x =x = 124 

A B D E F C Exterior angles A B D E F C Exterior Angle of a Cyclic Quadrilateral Now, let us review what exterior angles of a triangle are.

In the figure, ABCD is a cyclic quadrilateral. A B D C E When BC is produced to E,  DCE is called an exterior angle of ABCD.  A is called the interior opposite angle corresponding to  DCE. Exterior Angle of a Cyclic Quadrilateral

A B D C  BCF is an exterior angle of ABCD.  A is the interior opposite angle corresponding to  BCF. Exterior Angle of a Cyclic Quadrilateral F Similarly, when DC is produced to F,

A B D C E Let us try to find the relationship between x and y. x y  BCD = 180  – x y = 180  –  BCD = 180  – (180  – x) ∴ y = x opp.  s, cyclic quad. adj.  s on st. line

A B D C E Theorem 1.19 The exterior angle of a cyclic quadrilateral is equal to its interior opposite angle. Abbreviation: ext. , cyclic quad.  DCE =  A

120  x 27  Example: Find x in the figure. A B D C E DCE is a straight line. ext. , cyclic quad.  sum of △ x In △ ABD,

Follow-up question Find x and y in the figure. A B C 70  x D E 30  y x ADE and BCE are straight lines. In △ CED, y = x + 30  ext. , cyclic quad. ext.  of △

14 September 2015Ronald HUI

14 September 2015Ronald HUI Chapter 1 SQ1: 2/10 (Fri) SQ1: 2/10 (Fri) Revision Ex: 30/9 (Wed) Revision Ex: 30/9 (Wed) Time to work harder please!!! Time to work harder please!!!