As we know that D is the midpoint and angle y is formed within A and midpoint D it must be larger than x.
To see this think about two triangles formed out of ABC (one with angle x as the lower left angle and one with angle y as lower left angle).
In both triangles, angle ACB will be the same.
Angle ABC however will be smaller for triangle with angle y than for triangle with angle x.
Knowing that the sum of the angles of a triangle must be 180°, while angle ACB stays the same, y° must be larger than x° (180° = y° + ACB° + fraction of ABC° & 180° = x° + ACB° + ABC°) -> Answer B is correct

the farther we will move from the mid point the smaller Y-angle will be. We find here X angle in the farthest point. So, angle Y must be> angle X

We can compare the angles of the large triangle and the smaller triangle that includes the angle with a measure of y degrees.

We also need to remember that the sum of the measures of the internal angles of any triangle will be 180 degrees, and therefore those sums will be equal for any triangles.

The larger triangle is made up of these three angles: angle ACB, the angle that has a measure of x degrees, and the full angle ABC.

The smaller triangle is made up of these three angles: angle ACB, the angle that has a measure of y degrees, and an angle that has a smaller measure than angle ABC (since it is part of the larger angle). That third angle can be consider to have a measure equal to the measure of ABC minus a positive value (we'll call that z).

\(ACB + x + ABC = ACB + y + (ABC - z)\)
\(x = y - z\)
Therefore, since z is positive, it must be the case that:
\(x < y\)

Interestingly enough, the piece of information that D is the midpoint of AC is of no value in answering the question.

The answer is B) y is bigger.

Here is how I went about it.
There are two triangles in the picture (one with angle y and another with angle x)
We know that the sum of angles in a triangle is 180 degrees -----> (1)

Let b be the angle with vertex at B for the triangle that has the angle x
Let b' be the angle with vertex at B for the triangle that has the angle y
Let c be the angle with vertex at C.

Now, from (1)

b + x+ c = 180 => b + x = 180 - c ------> (2)

b' + y + c = 180 => b' + y = 180 - c - ----> (3)

Equating (2) and (3) b' + y = b + x ---------> (4)

But b' < b from diagram

Means that for (4) to be satisfied, y > x

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