Question

CDE IS AN EQUILATERAL TRIANGLE FORMED ON A SIDE CD OF A SQUARE ABCD. SHOW THAT ∆ADE CONGRUENT TO ∆BCE

Answer 1

Triangles ADE and BCE have two congruent sides, and the angle between the sides is congruent as well. Therefore, the triangles are congruent.

1. We have AD = BC because they are diagonals of a square
2. We have DE = CE because they are sides of an equilateral triangle
3. Angles BCE and ADE are congruent because we can write them as:

`BCE = BCD+DCE,\quad ADE = ADC + CDE`

And we have BCD = ADC because they are both 45° angles, because the diagonals of a square are also bisectors

Also, we have CDE = DCE because they are both 60° angles, because they are angles of an equilateral triangle.

So, BCE and ADE are composed by the same pieces, and are therefore congruent.