The problem describes a triangle ABC with a smaller equilateral triangle DEC inside it. We are given that angle BAC is $50^{\circ}$ and we need to find the size of angle ABC.

1. Problem Description

The problem describes a triangle ABC with a smaller equilateral triangle DEC inside it. We are given that angle BAC is $50^{\circ}$ and we need to find the size of angle ABC.

2. Solution Steps

First, we know that triangle DEC is equilateral, which means all its angles are equal to $60^{\circ}$. Thus, angle DCE = $60^{\circ}$.

Next, consider the triangle ABC. The sum of the angles in a triangle is $180^{\circ}$. Therefore:

$angle BAC + angle ABC + angle ACB = 180^{\circ}$

We know that $angle BAC = 50^{\circ}$. We also know that $angle ACB$ is adjacent to $angle DCE$, so $angle ACB + angle DCE = angle BCD + angle DCA$. These are angles on a straight line therefore add up to $180$. We can describe $angle ACB = angle ACE + angle ECB$

$angle ECB + angle DCE=180 - 60 = 120$ degrees. Now because B, C, and D are on a straight line we now know that angle ACB equals 180 degrees so:

Since points D, C, and B are collinear, they form a straight line. Therefore, the angle DCB is a straight angle, which is $180^{\circ}$.

$angle ACB + angle DCE = 180^{\circ}$

$angle ACE + angle ECB + 60 = 180^{\circ}$

This line isn't helping us so we go back to our original equation

$angle BAC + angle ABC + angle ACB = 180^{\circ}$

We know $angle BAC$ =

5

0. Because $BCD$ is a straight line, $angle ACB = 180 - angle DCE$

Since $angle DCE = 60$, then $angle ACB = 180 - 60 = 120 - angle ECA$. Thus we only know $angle ACB$

Therefore:

$50 + angle ABC + angle ACB = 180$

If we label angle ABC with x:

$x + 50 + (angle ACB) = 180$

$angle ECB+60= 180$ so angles B and C are equal and the only degree difference will be 60 degrees in total.

Since triangle EDC is equilateral, angles EDC, DCE, and CED equal $60^{\circ}$.

$angle ACB=180 - 60 =120^{\circ}$ if we are calculating straight from D,C, B straight line. We also have:

$50 + angle ABC + angle ACB = 180$

$50 + angle ABC +120-60= 180$.

$110-60 = 180$ is clearly not correct and my working is flawed.

Given the information, there's not enough to solve for angle ABC. Because Triangle ABC is not equilateral

There should be a ratio given to solve this properly. If we were to guess, then the angle would be $65$.

Sum of angles in triangle ABC: $angle BAC + angle ABC + angle ACB = 180^{\circ}$

Since angle BAC = 50, we have: $50 + angle ABC + angle ACB = 180^{\circ}$

$angle ABC + angle ACB = 130^{\circ}$

Without further information, we can't determine the exact value of angle ABC.

3. Final Answer

Not enough information. The value cannot be determined.