To solve the problem, let's first consider the properties of the triangles involved. The given problem states that $\angle ABC = \angle ADC$. This implies that the two triangles $\triangle ABC$ and $\triangle ADC$ share the same angle $\angle ABC$, and they also share $\angle A$ in $\triangle ABC$ and $\triangle ADC$.

Step 1: Identify Similar Triangles

The triangles $\triangle ABC$ and $\triangle ADC$ are similar by the Angle-Angle (AA) similarity criterion, as they both have two pairs of corresponding angles equal.

Step 2: Apply Similarity Ratios

Since the triangles are similar, the corresponding sides are proportional. This means: $\frac{AB}{AD} = \frac{BC}{DC} = \frac{AC}{AC}$

Step 3: Given Data and Proportionality

• $AB = 16$ cm
• $BC = 20$ cm
• $AC = 24$ cm

Let $AD = x$ and $BD = y$. Since $\angle ABC = \angle ADC$ and triangles are similar: $\frac{AB}{AD} = \frac{AC}{DC}$

Step 4: Use Pythagoras and Ratios

Now, we use the fact that $AC = AD + DC$: $AC = x + y = 24 \text{ cm}$ Since $AB$ and $AD$ are in the same proportion: $\frac{16}{x} = \frac{24}{y}$

Now solving for $y$ in terms of $x$: $y = \frac{3x}{2}$

Substitute this back into the equation for $AC$: $x + \frac{3x}{2} = 24$ $\frac{5x}{2} = 24$ $5x = 48$ $x = 9.6 \text{ cm}$

Step 5: Find $y$:

$y = \frac{3x}{2} = \frac{3 \times 9.6}{2} = 14.4 \text{ cm}$

Final Answer:

• $AD = 9.6$ cm
• $BD = 14.4$ cm

Would you like more details or explanations? Here are some related questions you might be interested in:

1. What is the Angle-Angle (AA) similarity criterion?
2. How do you apply the concept of proportionality in similar triangles?
3. Can you derive the formula for finding a missing side using triangle similarity?
4. What other similarity criteria exist besides AA?
5. How can trigonometry help in finding angles and sides in non-right triangles?
6. What are the properties of special triangles like 30-60-90 or 45-45-90 triangles?
7. What is the significance of the Pythagorean theorem in triangle problems?
8. How does the concept of similarity apply in geometry beyond triangles?

Tip: When dealing with similar triangles, always ensure that the corresponding sides and angles are properly matched to avoid any errors in calculation.