To find the measure of angle BCD, we start with the information that angles ACB and BCD are supplementary, meaning their measures add up to 180 degrees.
Given:
- Measure of angle ACB = \(5x\) degrees
- Measure of angle BCD = \(2x + 5\) degrees
Setting up the equation based on the supplementary angles: \[ 5x + (2x + 5) = 180 \]
Now, we can simplify the equation: \[ 5x + 2x + 5 = 180 \] \[ 7x + 5 = 180 \]
Next, we will isolate \(x\): \[ 7x = 180 - 5 \] \[ 7x = 175 \] \[ x = \frac{175}{7} \] \[ x = 25 \]
Now, we can find the measure of angle BCD by substituting \(x\) back into its expression: \[ \text{Measure of angle BCD} = 2x + 5 \] \[ = 2(25) + 5 \] \[ = 50 + 5 \] \[ = 55 \text{ degrees} \]
Thus, the measure of angle BCD is \(\boxed{55}\) degrees.