□ ABCD is a trapezium, AB || CD, diagonals of trapezium intersects in point P.Write the answers of the following questions :(a) Draw the figure using given information.(b) Write any one pair of alternate angles and opposite angles.(c) Write the names…

Answer: (b) alternate angles ∠PAB and ∠PCD (also ∠PBA and ∠PDC); vertically opposite angles ∠APB and ∠CPD. (c) Δ PAB ~ Δ PCD, by the AA test of similarity.

(a) Draw seg AB and seg CD parallel to each other, with AB longer than CD; join A to C and B to D. The diagonals AC and BD intersect at P.

(b) Since AB || CD and AC is a transversal, the alternate angles are ∠PAB and ∠PCD. The vertically opposite angles are ∠APB and ∠CPD (also ∠APD and ∠BPC).

(c) In Δ PAB and Δ PCD:

∠PAB ≅ ∠PCD (alternate angles, AB || CD)

∠PBA ≅ ∠PDC (alternate angles, AB || CD)

∴ Δ PAB ~ Δ PCD (AA test of similarity).