To solve the problem, let's break it down into steps:

Step 1: Identify Similar Triangles

Given that $\angle ABC = \angle ADC$, we can say that triangles $\triangle ABC$ and $\triangle ADC$ are similar by the AA similarity criterion (since they share angle $\angle A$, and $\angle ABC = \angle ADC$ is given).

Step 2: Set Up Proportions for Similar Triangles

Since $\triangle ABC \sim \triangle ADC$, the corresponding sides of these triangles are proportional:

$\frac{AB}{AD} = \frac{BC}{DC} = \frac{AC}{AC}$

Step 3: Use Known Values to Find $AD$ and $BD$

Let's use the given side lengths: $AB = 16$ cm, $BC = 20$ cm, and $AC = 24$ cm.

From the proportionality:

$\frac{AB}{AD} = \frac{AC}{AC} \implies \frac{16}{AD} = \frac{24}{AC}$

Since the corresponding sides $AB$ and $AD$ relate to $AC$ (a common side), the proportion simplifies to:

$\frac{16}{AD} = \frac{24}{AC}$

However, without knowing $AC$, we don't have enough information directly. Instead, we can find the lengths by applying the cosine rule or additional information if available.

Given $AB$ and $BC$ lengths, if $BD = x$, $AD = 16 - x$, we could set up:

$x + AD = 20 \text{ cm (length of BC)}$

And solve the system:

If not, you would typically use additional geometry tools or context, such as a given length for AD directly or related through trigonometry to work out unknowns.

Would you like a more detailed solution, or need a breakdown of steps for each length?

Related Questions:

1. What are the main properties of similar triangles?
2. How can the AA similarity criterion be used to prove triangles are similar?
3. What other pairs of angles could create similar triangles in this figure?
4. How would you calculate side lengths if only one angle and one side were known?
5. Can the cosine rule help in this situation, and how?
6. If the angles were slightly different, what would change in the approach?
7. How do corresponding sides in similar triangles relate mathematically?
8. What happens if the given sides were in a different ratio?

Tip: Always verify your similar triangles by checking multiple side ratios or angles to ensure they correspond correctly.