The problem states that triangle $DEC$ is an equilateral triangle and angle $BAC$ is $50$ degrees. The task is to find the size of angle $ABC$.

1. Problem Description

The problem states that triangle $DEC$ is an equilateral triangle and angle $BAC$ is $50$ degrees. The task is to find the size of angle $ABC$.

2. Solution Steps

Since triangle $DEC$ is equilateral, all its angles are $60$ degrees. Therefore, angle $DCE = 60$ degrees.

Angles $ACB$ and $DCE$ form a straight line, so they are supplementary angles, meaning their sum is $180$ degrees. Thus,

$ACB + DCE = 180$.

$ACB + 60 = 180$.

$ACB = 180 - 60 = 120$ degrees.

The sum of the angles in triangle $ABC$ is $180$ degrees. Therefore,

$BAC + ABC + ACB = 180$.

We know that $BAC = 50$ and $ACB = 120$.

$50 + ABC + 120 = 180$.

$ABC + 170 = 180$.

$ABC = 180 - 170 = 10$ degrees.

3. Final Answer

The size of angle $ABC$ is 10 degrees.