Question

In the given figure below, if \[\Delta ABE\cong \Delta ACD\] , prove that \[\Delta ADE\sim\Delta ABC\] .

Answer 1

Hint: To prove \[\Delta ADE\sim\Delta ABC\] , we first consider the congruence of \[\Delta ABE\text{ }\] and \[\Delta ACD\] . From this, we will get $\text{AB=AC}...\text{(i)}$ and \[\text{AD=AE}...\text{(ii)}\] . Now, divide equation (ii) by (i) to get \[\dfrac{\text{AD}}{\text{AB}}\text{=}\dfrac{\text{AE}}{\text{AC}}\] . In triangles \[\Delta ADE\text{ }\] and \[\Delta ABC\] , \[\angle A\] is common. Hence the similarity between these triangles can be proved using the SAS theorem.

Complete step by step answer:
We have to prove that \[\Delta ADE\sim\Delta ABC\] . It is given that \[\Delta ABE\cong \Delta ACD\] .
Let us first analyze the congruency, that is, \[\Delta ABE\cong \Delta ACD\] .
Hence, using Corresponding parts of Congruent triangles (CPCT), we get
$\text{AB=AC}...\text{(i)}$
\[\text{AE=AD}\]
\[\Rightarrow \text{AD=AE}...\text{(ii)}\]
Now, let us divide equation (ii) by (i). We will get
\[\dfrac{\text{AD}}{\text{AB}}\text{=}\dfrac{\text{AE}}{\text{AC}}...(\text{iii)}\]
Now, let us prove \[\Delta ADE\sim\Delta ABC\] .
From \[\Delta ADE\sim\Delta ABC\] ,
\[\angle A=\angle A\] (Common angle)
From (iii), we got,
\[\dfrac{\text{AD}}{\text{AB}}\text{=}\dfrac{\text{AE}}{\text{AC}}\]
We know that if the two sides of a triangle are in the same proportion of the two sides of another triangle, and the angle inscribed by the two sides in both the triangles are equal, then two triangles are said to be similar. This is the Side-Angle-Side(SAS) similarity rule.
Hence, by SAS rule,
\[\Delta ADE\sim\Delta ABC\]
Hence proved.

Note: There are three ways to prove the similarity between two triangles. The first one is Angle-Angle Similarity (AA) which states that if any two angles of a triangle are equal to any two angles of another triangle, then the two triangles are similar to each other. In \[\Delta ADE\text{ }\] and \[\Delta ABC\] only \[\angle A\] comes as common. Hence, we did not use this property. The second property is Side-Angle-Side(SAS) similarity that we have done in this question. The last one is SSS or Side-Side-Side Similarity which states that if all the three sides of a triangle are in proportion to the three sides of another triangle, then the two triangles are similar. In \[\Delta ADE\text{ }\] and \[\Delta ABC\] , it is clear that side DE and BC cannot be in proportion. Hence, we did not use this rule.