Given: angle DEB & DCA are congruent, DA & DB are congruent, Angle A & B are congruent Prove:AC & BE congruent

Question

this proof has two equilateral triangles one triangle is DCA and the other triangle is DEB

Answer 1

hi
the details which you gave have the answer for this question.

angle E= angle C and angle A= angle B and DA=DB

from this given data i will not assume that the triangles are equilateral, I will prove they are.

calculating the area of the 2 triangles with the formula (1/2)*base*height we can prove these 2 triangles are congruent and AC=BE

since the angle E=angle C and DA=DB then DB and DA must be the sides opposite these angles E and C respectively.
hence the line drawn(at angle of 90 degrees to base DB) from angle E to DB and line drawn(at and angle of 90 degrees to DA) from angle C to DA must be of equal magnitude (otherwise the angles cant be equal and/or sides cant be equal) ,

hence the length of this line drawn which is called as the height of the triangle is proved to be equal

now calculating the area with above formula (1/2)*base*height
heights of 2 triangles are proved to be equal equal and bases are
given to be equal hence areas of these 2 triangles must be equal.

when area of 2 trianlges are equal and 2 sets of angles are equal and one side is equal then the 2 triangles must be absolutely congruent. hence AC must be equal to BE