Question

For the given trapezoid ABCD, the triangles DAB and CBA are congruent, which of the following statements is true?

A). \[\left( a \right)BD\cong CD\]
B). \[\left( b \right)DA\cong DC\]
C). \[\left( c \right)\angle BDE\cong \angle DAB\]
D). \[\left( d \right)\angle ACB\cong \angle BDA\]

Answer 1

Hint: To solve this question, first of all, define the congruence of two triangles, how it relates angles and sides of two triangles and what is the exact form of it. After defining, consider all the given options and check the options that are only dealing with the angles and sides that contradict the options which are used to get the result.

Complete step-by-step solution
Given that the trapezoid ABCD as

The triangle DAB and triangle CBA are congruent. Let us first define the congruent triangles. Two triangles are called congruent if they have exactly the same three sides and the same three angles. The equal sides and angles may not be in the same position. Also, the corresponding sides and the angles of the congruent triangles are equal/congruent.
As triangle DAB is congruent to triangle CBA, the sides are congruent.
\[\Rightarrow AD\cong BC\]
\[\Rightarrow AC\cong BD\]
So, the options (a) and (c) are wrong.
Consider the angles now. In triangle DAB and CBA,
\[\angle BAD\cong \angle ABC\left[ \text{As }\Delta DAB\cong \Delta CBA\Rightarrow \text{angles congruent} \right]\]
\[\angle ADB\cong \angle ACB\left[ \text{As }\Delta DAB\cong \Delta CBA\Rightarrow \text{angles congruent} \right]\]
\[\angle ABD\cong \angle BAC\left[ \text{As }\Delta DAB\cong \Delta CBA\Rightarrow \text{angles congruent} \right]\]
So, matching from options, we see that, \[\angle ACB\cong \angle BDA\] is the right answer. Hence, option (d) is the right answer.

Note: Do not confuse while considering angle BDA and angle ADB.

Whether it be angle BDA or ADB, they are the same as represented in x as in the figure.
\[\begin{align}
  & \Rightarrow \angle ACB\cong \angle ADB \\
 & \Rightarrow \angle ACB\cong \angle BDA \\
\end{align}\]