Question

ABCD is a trapezium in which AB parallel CD and AD =BC (see the given figure).

Identify the incorrect statements from the following.

• A.
  $\angle A=\angle B$
• B.
  $\angle C=\angle D$
• C.
  $\Delta ABC\cong \Delta BAD$
• D.
  AC ≠ BD

Answer 1

The correct option is D

AC ≠ BD
Let us extend AB. Then, draw a line through C, which is parallel to AD, intersecting AE at point E. It is clear that AECD is a parallelogram.
AD = CE (Opposite sides of parallelogram AECD)
However, AD = BC (Given)
Therefore, BC = CE
$\angle CEB=\angle CBE$
$\angle CBE+\angle CBA={180}^{\circ }$ Linear pair)………….(1)
$\angle CBA+\angle DAB=180$ (Angles on the same side of transversal)……….(2)
From (1) and (2) $\angle A=\angle B$

AB || CD.
$\angle A+\angle D={180}^{\circ }$ ( Angles on the same side of the transversal).
Also, $\angle C+\angle B={180}^{\circ }$ ( Angles on the same side of the transversal).
$\angle A+\angle D=\angle C+\angle B$.
However, $\angle A=\angle B$ [Using the result obtained in (i)].
$\therefore \angle C=\angle D$.

In $\Delta ABC$ and $\Delta ABD$,
AB = BA (Common side)
BC = AD (Given)
$\angle B=\angle A$ (Proved before)
$\Delta ABC\cong \Delta BAD$ (SAS congruence rule).

Hence , AC = BD [CPCT].