Question

In the given figure, if \[\vartriangle ABE \cong \vartriangle ACD\], show that \[\vartriangle ADE \sim \vartriangle ABC\].

Answer 1

Hint: Use CPCT on the given congruent triangles and write equal sides of two triangles. We prove two triangles similar by the SAS theorem of similarity. Use the proportional sides and common angle to show triangles similar.
CPCT theorem: CPCT stands for corresponding parts of congruent triangles. Theorem states that when two triangles are congruent, we can write the corresponding parts of two triangles equal.
SAS theorem of similarity states that if two sides in one triangle are proportional to two sides in another triangle and the included angle in both are equal, then the two triangles are similar.

Complete step-by-step solution:
We are given that \[\vartriangle ABE \cong \vartriangle ACD\]
When two triangles are given congruent to each other, then by CPCT the corresponding sides and corresponding angles of both triangles are congruent to each other.
\[ \Rightarrow AB = AC,BE = CD,AE = AD\] are corresponding equal sides of two triangles.
\[ \Rightarrow \angle ABE = \angle ACD,\angle BAE = \angle CAD,\angle AEB = \angle ADC\] are corresponding equal angles of two triangles.
Since we have \[AB = AC,AE = AD\]
Then we can write the fraction \[\dfrac{{AD}}{{AB}} = \dfrac{{AE}}{{AC}}\]...............… (1)
We will prove the two triangles \[\vartriangle ADE,\vartriangle ABC\] similar to each other.
In\[\vartriangle ADE,\vartriangle ABC\];
\[\dfrac{{AD}}{{AB}} = \dfrac{{AE}}{{AC}}\] {From (1)}
\[\angle BAE = \angle CAD\] {Common angle}
We have two sides in one triangle that are proportional to two sides in another triangle and included angles equal in both the triangles.
Then by SAS similarity theorem, we can say that\[\vartriangle ADE \sim \vartriangle ABC\]
\[\therefore \vartriangle ADE \sim \vartriangle ABC\]

Note: Students many times confuse SAS similarity theorem with SAS congruence. They try to prove sides of triangles congruent and equal, but keep in mind here we have to prove triangles similar and the rules are different for proving congruence and different for proving similarity. Also, focus on the pair of equal angles in both triangles as they should be between the proportional sides, i.e. the angle should be included and not any of the angles.