Theory question 111
Angles and Polygons(a) In the diagram, AB//CD and BC//FE. CD̂E = 75° and DÊF = 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 angle DAO = 35°. Find: (i) ∠ABC; (ii) ∠ADC.
(c) Using the diagram in 11(b) above, calculate, correct to 3 significant figures, the length of: (i) arc ABC; (ii) the chord AD. [Take π = 3.142]
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(a) Through D draw a line parallel to BC and FE. Its angle with DE = 26° (alternate), so its angle with DC = 75° − 26° = 49°. Hence ∠BCD = 49° (alternate) and x = ∠ABC = ∠BCD = 49° (AB ∥ CD).
y = 360° − 49° = 311°.
(b) ∠AOC (non-reflex) = 360° − 190° = 170°.
(i) ∠ABC = ½ × 190° = 95°. (ii) ∠ADC = ½ × 170° = 85°.
(c)(i) Arc ABC = 170/360 × 2 × 3.142 × 7 ≈ 20.8 cm.
(ii) ΔAOD is isosceles with ∠OAD = ∠ODA = 35°, so ∠AOD = 110°. AD = 2 × 7 × sin 55° ≈ 11.5 cm.
An angle at the circumference is half the angle at the centre on the same arc.