(a) 
In the diagram, AB // CD and BC // FE. \(\stackrel\frown{CDE} = 75°\) and \(\stackrel\frown{DEF} = 26°\). Find the angles marked x and y.
(b) 
The diagram shows a circle ABCD with centre O and radius 7 cm. The reflex angle AOC = 190° and < DAO = 35°. Find :
(i) < ABC ; (ii) < ADC.
(c) Using the diagram in (b) above, calculate, correct to 3 significant figures, the length of : (i) arc ABC ; (ii) the chord AD. [Take \(\pi = 3.142\)].
Explanation
(a) 
105° + 26° + x = 180°
131° + x = 180°
x = 180° - 131°
= 49°.
131° + x + 131° + 360 - y = 360°
131° + 49° + 131° + 360 - y = 360°
671° - y = 360°
y = 671° - 360°
= 311°.
(b)(i) \(\stackrel\frown{ABC} = \frac{1}{2} \stackrel\frown{AOC} reflex\)
= \(\frac{1}{2} \times 190° = 95°\)
(ii) \(\stackrel\frown{ADC} = \frac{1}{2} \stackrel\frown{AOC}\)
= \(\frac{1}{2} (360° - 190°)\)
= \(\frac{1}{2} \times 170°\)
= 85°
(c) (i) \(\stackrel\frown{ABC} = \frac{360 - 190}{360} \times 2\pi r\)
= \(\frac{170}{360} \times 2 \times 3.142 \times 7\)
= \(20.77 cm\)
\(\approxeq 20.8 cm\) (3 significant figures)
(ii) \(AD = 2 \times 7\cos 35\)
= \(14 \times 0.8192\)
= \(11.468 cm \approxeq 11.5 cm\) (3 significant figures)