Question

In the given figure, $\Delta CDE$ is an equilateral triangle on a side CD of a square ABCD. Show that $\Delta ADE \cong \Delta BCE$.

Answer 1

Hint: The above question is based on the concept of congruent triangles.
Congruent triangles are the one which have all the three sides equal and all the three angles equal.
We will prove the congruence condition of the given triangles by examining the sides and angle criteria.

Complete step-by-step solution:
Let's explain the congruent triangle concept in detail first and then we will prove the two given triangles congruent.
Congruent triangles are the one which have all the three sides equal and all the three angles equal. Even if we flip the congruent triangles their sides and angles remain the same. In other words, if superimposition of sides and angles take place then the triangle is said to be congruent. When we witness that the angles of the triangle are equal then the triangles are not said to be congruent unless all the three sides are of the same length.
Now, we will prove the condition of congruency .
$ \Rightarrow \angle ADE = \angle BCD$ (angles equal to $90^{\circ}$)

$ \Rightarrow \angle CDE = \angle DCE$ (angles equal to $60 ^{\circ}$ )

On adding the two angles we get

$\therefore \angle ADE = \angle BCE$ (angles equal to $150^{\circ}$)
In triangle ADE and BCE
$ \Rightarrow \angle ADE = \angle BCE$ (angles equal to $150^{\circ}$)
$ \Rightarrow DE = CE$ (sides of equilateral triangle)
$ \Rightarrow AD = BC$ (sides of the square formed in the figure)
$\Delta ADE \cong \Delta BCE$

Thus, proved that triangle ADE and BCE are congruent

Note: A Congruent triangle has four types of conditions such as RHS ( Right angle hypotenuse side),SAS(side angle side), ASA(angle side angle), SSS(side side side) using these conditions according to the triangle given we will prove the triangle congruent. In the problem above we have used RHS congruence criterion.