(a) Evaluate, without using mathematical tables, \(17.57^{2} - 12.43^{2}\).
(b) Prove that angles in the same segment of a circle are equal.
(a) \(17.57^{2} - 12.43^{2}\)
Using the difference of two squares method,
= \((17.57 + 12.43)(17.57 - 12.43)\)
= \((30.00)(5.14)\)
= \(154.2\)
(b) 
Given: D and C are points on the major arc of circle ADCB. To prove that < ADB = < ACB.
Construction : Join A and B to O, the centre of the circle .
Proof: With the lettering \(< AOB = 2x_{1}\) (angle at the centre is twice that subtended at the circumference)
But \(< AOB = 2x_{2}\) (the same theorem applies here)
\(\therefore 2x_{1} = 2x_{2} \implies x_{1} = x_{2}\)
\(\therefore < ADB = < ACB\)
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