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HKDSE Mathematics Ronald Hui Tak Sun Secondary School
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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)
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14 September 2015Ronald HUI
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14 September 2015Ronald HUI
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14 September 2015Ronald HUI
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14 September 2015Ronald HUI
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Book 5A Chapter 1 Cyclic Quadrilaterals
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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
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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
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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
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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
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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
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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
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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 + 124 = 180 x =x = 124
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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.
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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
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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,
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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
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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
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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,
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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 △
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14 September 2015Ronald HUI
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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!!!
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