We know that the angle bisector of angle BAD divides it into two equal angles.
Let's denote the measure of angle BCA as "x".
Since the bisector of angle BAD divides it into two equal angles, we can say that angle CAD (which is equal to angle BAD) is also x.
Therefore, we have:
angle BAD = angle CAD = x
Now, we know that the sum of angles in a triangle is 180 degrees. So, we can write an equation:
angle BCA + angle CAD + angle BAD = 180 degrees
Substituting the known values:
x + x + 50 degrees = 180 degrees
2x + 50 degrees = 180 degrees
Subtracting 50 degrees from both sides:
2x = 180 degrees - 50 degrees
2x = 130 degrees
Dividing both sides by 2:
x = 130 degrees / 2
x = 65 degrees
Therefore, the measure of angle BCA is 65 degrees.