1. Problem: Calculate sizes of angles in polygons and circles given various conditions. 2. (i) In a regular hexagon, central angle between two adjacent vertices (e.g., ∠AOB) is given by dividing 360° by number of sides: $$\angle AOB=\frac{360^\circ}{6}=60^\circ.$$ (ii) For the regular octagon, each side subtends a central angle of $$\frac{360^\circ}{8}=45^\circ.$$ Since AC is a side of the octagon, chord AC subtends angle at circumference ∠ACB, which is half the central angle subtended by arc AB: $$\angle ACB=\frac{1}{2}\times 60^\circ=30^\circ.$$ (iii) For ∠ABC, consider that in a circle angles subtended by the same chord are equal, so using properties of regular polygons and circle geometry, we conclude $$\angle ABC=45^\circ.$$ 3. In regular pentagon ABCDE, all central angles are $$\frac{360^\circ}{5}=72^\circ.$$ Angles at circumference subtended by chords can be found. Ratio $$\frac{\angle EDA}{\angle ADC} = \frac{\text{arc } EA}{\text{arc } DC} = \frac{72^\circ}{72^\circ} = 1.$$ 4. Given AB = BC = CD and ∠ABC = 132°, isosceles triangle properties apply: (i) ∠AEB is an exterior angle related to ∠ABC and ∠BCD, compute using circle theorems. (ii) ∠AED depends on arc lengths, calculated via central and inscribed angles. (iii) ∠COD at the center from chords same length equals $$360^\circ - 2 \times 132^\circ=96^\circ.$$ 5. Given arcs AB = 2 * arc BC and ∠AOB = 108°: (i) Using arc lengths and inscribed angle theorem, $$\angle CAB=\frac{1}{2} \times \text{arc } CB=\frac{1}{2} \times 36^\circ=18^\circ.$$ (ii) $$\angle ADB = \frac{1}{2} \times \text{arc } AB=54^\circ.$$ 6. For regular pentagon side AB and hexagon side AC, triangle ABC angles are $$\angle A = 54^\circ, \ \angle B=72^\circ, \ \angle C=54^\circ$$ (using polygon internal angle formulas and circle geometry). 7. Given BD side of regular hexagon, DC side of regular pentagon, and AD diameter: (i) $$\angle ADC = 90^\circ$$ (angle subtended by diameter), (ii) $$\angle BDA = 60^\circ,$$ (iii) $$\angle ABC = 30^\circ,$$ (iv) $$\angle AEC = 30^\circ.$$ 8. TEST YOURSELF answers: (a) (iii) 360°; (b) (iv) x°= y°=45°; (c) (iii) 128°; (d) (ii) 66°; (e) (iv) 160°; (f) True as given. These answers use polygon and circle angle properties including central angles, inscribed angles, and arcs subtended by chords.