Question

\( ABCD \) is a trapezium, \( AB \parallel CD \), and the diagonals of the trapezium intersect at point \( P \). Write the answers to the following questions:
[(a)] Draw the figure using the given information.
[(b)] Write any one pair of alternate angles and opposite angles.
[(c)] Write the names of similar triangles with a test of similarity.

Hint:

In a trapezium, diagonals often form similar triangles by the AA similarity criterion due to parallel sides and corresponding angles.

Answer 1

(a) Drawing the figure:
The figure is shown above, where \( AB \parallel CD \), and the diagonals \( AC \) and \( BD \) intersect at \( P \).
(b) Alternate and opposite angles:
One pair of alternate angles: \[ \angle APD = \angle CPB. \] One pair of opposite angles: \[ \angle APB = \angle CPD. \] (c) Names of similar triangles:
The triangles \( \triangle APB \) and \( \triangle CPD \) are similar by the AA similarity test: \[ \angle APB = \angle CPD \quad \text{and} \quad \angle PAB = \angle PCD. \]