A student is given the following measurements for a triangle: side AB=5 cm, side BC=4 cm and angle ∠A=30∘. Answer the following questions.
(i) Write a step-by-step ruler-and-compasses construction to obtain at least one triangle satisfying the given data.
(ii) Carry out the construction on paper and state whether the triangle obtained is unique. Give reasons for your observation.
(iii) Explain geometrically how the position of ∠A with respect to the given sides can lead to two distinct non-congruent triangles or to no triangle at all.
(iv) Using the same numerical data, specify a different arrangement of side(s) and angle that would guarantee exactly one triangle. Justify your answer by naming the relevant congruence condition.