n the figure angle abc=angle adc find 2 triangles in the figure which are similar to each other.If ab=16cm,bc=20cm and ac=24cm.find ad and bd

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.

Learn how to find similar triangles using the Angle-Angle (AA) similarity criterion. Solve for AD and BD given AB = 16 cm, BC = 20 cm, and AC = 24 cm in triangles ABC and ADC. Detailed step-by-step solution provided.

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