Question

Given, \[\Delta ABC\sim \Delta EDC.\] Find the condition that must be true according to the given information.

\[\left( \text{a} \right)\text{ }\overline{AE}||\overline{BD}\]
\[\left( \text{b} \right)\text{ }\overline{AE}\bot \overline{BD}\]
\[\left( \text{c} \right)\text{ }\overline{AB}||\overline{DE}\]
\[\left( \text{d} \right)\text{ }\overline{AB}\bot \overline{DE}\]

Answer 1

Hint: To solve the given question, we will first find out what similarity of two triangles is. Then, we will consider the triangles ABC and EDC. Then we will make use of the fact that the lines AE and BD intersect each other at C, so \[\angle BCA=\angle DCE\] because they are vertically opposite angles. Now, we will try to prove that the other two pairs of angles must also be equal for the triangles to be similar.

Complete step-by-step answer:
Before we solve the question, we must know about the similarity of two triangles. Two triangles are said to be similar if their corresponding angles are congruent and the corresponding sides are in proportion. In other words, similar triangles are the same shape but not necessarily the same size. Now, it is given in the question that triangle ABC and triangle EDC are similar. For the similarity of triangles, all the corresponding angles should be equal. Angles ACB and DCE will be equal because they are vertically opposite angles (lines AE and BD cross each other at C). Now, we have to prove two other pairs of these triangles as equal to angle ABC should be equal to angle CDE, and angle BAC should be equal to angle DEC. For these to be equal, the line AB and DE should be parallel because when AB || DE then angle BAC and angle DEC will be equal because they are alternate interior angles. Similarly, angle ABC and angle CDE will be equal because they are also alternate interior angles. Thus, AB and DE should be parallel, i.e. \[\overline{AB}||\overline{DE}.\]
Hence, the option (c) is the right answer.

Note: Here, we have assumed that ACE is a straight line and BCD is also a straight line. This assumption is necessary because then only angle ACB and angle DCE will be equal even if AB || DE but ACE and BCD are not straight lines, then also triangle ABC and triangle EDC will not be similar.