Question

In the given figure if \[AD = BC\] and \[AD\parallel BC\], then:

A. AB = AD
B. AB = DC
C. BC = CD
D. None

Answer 1

Hint: In this question, first of all prove that the triangles ADC and ABC are similar triangles by using SAS congruence rule. The corresponding sides in a similar triangle are in the same ratio. So, use this concept to reach the solution of the given problem.

Complete step-by-step answer:
Given that \[AD = BC\] and \[AD\parallel BC\].
Now, in \[\Delta ADC\] and \[\Delta ABC\] we have
\[
   \Rightarrow AD = BC{\text{ }}\left[ {\because {\text{given}}} \right] \\
   \Rightarrow \angle ADC = \angle ABC{\text{ }}\left[ {\because AD\parallel BC} \right] \\
   \Rightarrow AC = AC{\text{ }}\,{\text{ }}\left[ {\because {\text{common}}} \right] \\
\]
Therefore, by SAS congruence rule \[\Delta ADC \cong \Delta ADC\].
We know that the corresponding sides in similar triangle are in same ratio. So, we have
\[\dfrac{{AD}}{{BC}} = \dfrac{{DC}}{{AB}}\]
Since, \[AD = BC\] we have
\[
   \Rightarrow \dfrac{{AD}}{{AD}} = \dfrac{{DC}}{{AB}} \\
   \Rightarrow 1 = \dfrac{{DC}}{{AB}} \\
  \therefore AB = DC \\
\]
Thus, the correct option is B. AB = DC

Note: The SAS congruence rule states that two triangles are said to be similar if two sides and the included angle of one triangle are equal to two sides and included angle of another triangle. The symbol of similarity is ‘\[ \cong \]’.