To find the measure of angle ACD, we need to start by knowing more information about the properties of rhombus ABCD. Since CB equals 6, this means that opposite sides of the rhombus are congruent. Therefore, AD is also equal to 6.
Let's denote the measure of angle ADC as x. According to the given information, the measure of angle DAB is two times the measure of angle ADC, so we can say that the measure of angle DAB is 2x.
Since ABCD is a rhombus, the sum of the angles in a rhombus is always 360 degrees. In triangle ABC, we have the angles at A, B, and C. Since the opposite angles of a rhombus are congruent, angle ACD is equal to angle ABC, denoted as y.
Using the sum of the angles in triangle ABC:
y + 2x + y = 180
Combining like terms:
2y + 2x = 180
Dividing both sides by 2:
y + x = 90
Now we have a system of equations. We also know that CB and AD are congruent sides of the rhombus, so they are also congruent triangles.
CB = AD
6 = 6
Since triangles ACD and BDC share side DC, and also have congruent sides AC and CD, they are congruent triangles. This means that angle ACD is equal to angle BCD, denoted as z.
Using the sum of the angles in triangle BCD:
z + 2x + z = 180
Combining like terms:
2z + 2x = 180
Dividing both sides by 2:
z + x = 90
We now have a system of equations:
y + x = 90
z + x = 90
Subtracting x from both sides of both equations:
y = 90 - x
z = 90 - x
Since angle ACD is equal to angle BCD (z), we can substitute z in the first equation:
y = z
Now we have:
y = 90 - x
y = z
Therefore, angle ACD is equal to 90 - x degrees.