As shown in the figure below, points A, B, and D lie on a line. The measure of angle ABC (m∠ABC) is x°, and m∠CBD is (5x+4)°.

1. Break Down the Problem

We need to determine the relationship between the angles given that points A, B, and D are collinear, with angle ABC measuring $x^\circ$ and angle CBD measuring $(5x + 4)^\circ$. Since these angles are on a straight line, we know that:

$m∠ABC + m∠CBD = 180^\circ$

2. Relevant Concepts

The angles along a straight line add up to $180^\circ$. Therefore, we can set up the following equation based on the information given:

$x + (5x + 4) = 180$

3. Analysis and Detail

Now, we will simplify and solve the equation:

1. Combine like terms: $x + 5x + 4 = 180 \implies 6x + 4 = 180$

2. Subtract 4 from both sides: $6x = 180 - 4 \implies 6x = 176$

3. Divide by 6: $x = \frac{176}{6} \implies x = \frac{88}{3} \approx 29.33$

4. Verify and Summarize

Now we will verify our result by calculating $m∠CBD$:

$m∠CBD = 5 \left(\frac{88}{3}\right) + 4 = \frac{440}{3} + \frac{12}{3} = \frac{452}{3} \approx 150.67$

Now, check if the angles add up to $180^\circ$:

$m∠ABC + m∠CBD = \frac{88}{3} + \frac{452}{3} = \frac{540}{3} = 180$

Final Answer

Thus, the value of $x$ is $\frac{88}{3}$ or approximately $29.33^\circ$.