 Hello and welcome to the session. In this session, we discussed the following question which says, in the given figure, A B C D is a square and triangle E D C is an equilateral triangle, prove that A E is equal to B E, angle D A E is equal to 15 degrees. Let's move on to the solution. This is the figure given to us. We are given that A B C D is a square then triangle E D C is an equilateral triangle and we need to prove A E is equal to B E and angle D A E is equal to 15 degrees. We consider triangles A D E and B C E. In these triangles we have side A D of triangle A D E is equal to the side B C of triangle B C E since they are the sides of the square A B C D and we know that sides of the square are equal. Then we have the side D E of triangle A D E is equal to the side C E of triangle B C E since they are the sides of equilateral triangle E D C. Now since triangle E D C is equilateral triangle therefore all its angle would be of measure 60 degrees that is we have angle E D C is equal to angle E C D is equal to angle D E C is equal to 60 degrees. Then A B C D is a square so all its angles are of measure 90 degrees. So we have angle A is equal to angle B is equal to angle B C D is equal to angle C D A equal to 90 degrees. Consider this angle A D E angle A D E is equal to angle C D A plus angle E D C. Now angle C D A is of measure 90 degrees so this is equal to 90 degrees plus angle E D C which is of measure 60 degrees. So this total is equal to 150 degrees that is we have angle A D E is equal to 150 degrees. Now we consider the angle B C E angle B C E is equal to angle B C D plus angle E C D. Now we have angle B C D is of measure 90 degrees and angle E C D is of measure 60 degrees so we get angle B C E is equal to 90 degrees plus 60 degrees and this is equal to 150 degrees that is we now have angle A D E is equal to angle B C E. So now we have got n triangles A D E and B C E A D is equal to B C then D E is equal to C E and angle A D E is equal to angle B C E. Therefore we get triangle A D E is congruent to triangle B C E by S A S congruence rule. Now since both these triangles are congruent so their corresponding parts will be equal. So from here we get side A E of triangle A D E is equal to side B E of triangle B C E since they are the corresponding parts of congruent triangles that is C C D. So we have proved that A D E is equal to B E. Now consider triangle A D E in this angle A D E plus angle D E A plus angle E A D is equal to 180 degrees since we know that the sum of the angles of a triangle is 180 degrees. We know that D A is equal to D C as they are the sides of the square A B C D then D C is equal to D E as they are the sides of equilateral triangle E D C therefore we have D A would be equal to D E. Now as we have got D A equal to D E therefore this implies that angle D A E would be equal to angle D E A since we know that in a triangle angles opposite equal sides are equal. We can take both these angles equal to x degrees and putting this value of angle D A E and angle D E E in this we get angle A D E plus x degrees plus x degrees is equal to 180 degrees that is we now have angle A D E plus 2x degrees is equal to 180 degrees and we had already found angle A D E equal to 150 degrees so we put 150 degrees in place of angle A D E so from here we get 150 degrees plus 2x degrees is equal to 180 degrees that is we have 2x degrees is equal to 180 degrees minus 150 degrees equal to 30 degrees so from here we have x degrees is equal to 30 upon 2 degrees equal to 15 degrees we have taken angle D A E and angle D E A as x degrees so we have got angle D A E is equal to 15 degrees we were supposed to prove this so hence proved with this we complete the session hope you have understood the solution for this question.